Bernoulli Variables, Classical Exclusion Processes and Free Probability

Abstract

We present a new description of the known large deviation function of the classical symmetric simple exclusion process by exploiting its connection with the quantum symmetric simple exclusion processes and using tools from free probability. This may seem paradoxal as free probability usually deals with non-commutative probability while the simple exclusion process belongs to the realm of classical probability. On the way, we give a new formula for the free energy—alias the logarithm of the Laplace transform of the probability distribution—of correlated Bernoulli variables in terms of the set of their cumulants with non-coinciding indices. This latter result is obtained either by developing a combinatorial approach for cumulants of products of random variables or by borrowing techniques from Feynman graphs.

1 Introduction and Summary

The symmetric simple exclusion process (SSEP) [9, 10, 20] is an iconic model of out-of-equilibrium classical statistical physics [16, 25]. It describes particles on a line, hopping to the left and right but with the exclusion rule that two particles can never be at the same place. The SSEP played an important role in the emergence of the so-called macroscopic fluctuation theory [5, 6], which is a general, phenomenological framework, suited for dealing with diffusive out-of-equilibrium classical systems. A quantum version of the classical SSEP [2], named Q-SSEP, has recently been defined. Q-SSEP extends the SSEP in the sense that it keeps track of possible quantum mechanical interferences but in such a way that the classical SSEP is recovered when looking at the average behavior of quantum mechanical observables. It might play a role in a possible quantum extension of the classical macroscopic fluctuation theory [4].

Interestingly, free probability techniques play an important role in the study of the Q-SSEP, either in constructing its invariant measure [7] or in deciphering its dynamics [14]. Since the classical SSEP is embedded in the quantum SSEP, free probability also plays a role in understanding the known characteristics of SSEP, in particular its large deviation rate function. This may sound surprising as free probability has been introduced in the mathematical literature [8, 21, 24, 27] in order to deal with non-commutative probability while SSEP belongs to the realm of classical probability. The purpose of this manuscript is to explain this hidden role.

On the way, we solve the following problem, apparently simple but for which we did not find an answer in the literature and which reveals nice connections with combinatorial structures. Let $b_i$, $i=1,\cdots ,N$, be a collection of N Bernoulli variables, $b_i=0$ or 1 and let $K_n(b_{i_1},\cdots ,b_{i_n})$ be their cumulants. We call non-coincident these cumulants when the indices $i_1,\cdots ,i_n$ are pairwise distincts (i.e., there are no $p\not =q$ such that $i_p=i_q$). Since $b_i^2=b_i$ for all i, all other cumulants, and hence the joint distribution, are determined from the non-coincident cumulants. Let $Z[h]:={\mathbb {E}}\left[ e^{\sum _i h_i b_i}\right]$, be the Laplace transform of the joint distribution of the $b_i$’s. To make contact with physics terminology, we shall call it the partition function. It is clearly fully determined by the non-coincident cumulants, since

$$\begin{aligned} Z[h] = {\mathbb {E}}\left[ \prod _i(1 + b_i\,e_i)\right] , \end{aligned}$$

(1)

with $e_i:=e^{h_i}-1$.

The question is then: How to compactly write $W[h]:=\log Z[h]$, the generating function of the cumulants, including coincident indices, in terms of the non-coincident cumulants?

Of course, the answer to this question is easy when these variables are independent, since then the generating function factorizes, $Z_\textrm{free}[h]=\prod _i[1+g_i\, e_i]$, with $g_i:={\mathbb {E}}[b_i]$ the mean of $b_i$, and

$$\begin{aligned} W_\textrm{free}[h]=\sum _i \log \left[ 1+g_i e_i\right] . \end{aligned}$$

(2)

It informs on cumulants at coincident points, say at order two ${\mathbb {E}}[b_i^2]^c=K_2(b_i,b_i) =g_i(1-g_i)$ or three ${\mathbb {E}}[b_i^3]^c=K_3(b_i,b_i,b_i)=g_i - 3g_i^2 +2 g_i^3$, and similarly at higher orders. To later make contact with large deviation rate functions, let S[n] be the Legendre transform of W[h], that is: $S_\textrm{free}[n]=\max _{\{h_i\}}\left[ \sum _i h_in_i\right. \left. -W_\textrm{free}[h]\right]$, then

$$\begin{aligned} S_\textrm{free}[n] = \sum _i \left[ n_i\log \left( \frac{n_i}{g_i}\right) + (1-n_i)\log \left( \frac{1-n_i}{1-g_i}\right) \right] . \end{aligned}$$

(3)

A simple formula such as (2) does not hold in the correlated case. Nevertheless, as explained in Sect. 2, W[h] admits a representation as a sum over bipartite graphs whose weights are determined by the non-coincident cumulants.

$$\begin{aligned} W[h] = \sum _H\frac{\mu (H^\bullet )}{|\text {Aut}\, H|}\sum _{{{\mathcal {L}}}\in Lab(H)}w({{\mathcal {L}}}), \end{aligned}$$

(4)

where the sum is over all connected bipartite graphs H with an arbitrary number of black but at most N white vertices, and ${\mathcal {L}}$ denotes a labeling of the white vertices by distinct integer indices in [1, N]. To such a labeling ${\mathcal {L}}$ is associated a weight $w({\mathcal {L}})$ described below, see equations (35,36).

Things simplify in the large N scaling limit if we assume that the non-coincident cumulants scale in a specific way at large N. Namely, let us assume that

$$\begin{aligned} K_n(b_{i_1},\cdots ,b_{i_n}) = \frac{1}{N^{n-1}} \, \psi _n\left( \frac{i_1}{N},\cdots , \frac{i_n}{N}\right) \, \left( 1 + O\left( \frac{1}{N}\right) \right) , \end{aligned}$$

(5)

for $\psi _n(x_1,\cdots ,x_n)$ a collection of continuous functions and $h_i=h(\frac{i}{N})$ for h(s) a continuous function, then only trees contribute to the graph expansion of the cumulant generating functions and $W[h]\sim NF[h]$ at large N. In accordance with physics terminology, we shall call F[h] the free energy (per unit of volume). As explained in Sects. 2 and 3, the latter can be determined by solving an extremization problem :

$$\begin{aligned} F[h]= \max _{g(\cdot ); q(\cdot )} \left[ \int _0^1 \!\!\! \text {d}x\left[ \log \left( 1+g(x)e(x)\right) - q(x)g(x)\right] + F_0[q]\right] , \end{aligned}$$

(6)

with $e(x):=e^{h(x)}-1$ and $F_0[q]$ the generating function of non-coincident cumulants,

$$\begin{aligned} F_0[q]:= \sum _{n\ge 1}\frac{1}{n!}\int _0^1\! \psi _n(x_1,\cdots ,x_n) \prod _{k=1}^n q(x_k)\text {d}x_k . \end{aligned}$$

(7)

The extremization problem (6) has to be solved over all functions g and q on [0, 1], without specified boundary conditions. Comparing with the free formula (2), we observe that F[h] is given by a mean field like formula—the first term $\int \text {d}x \log \big (1+g(x)e(x)\big )$—with effective local density g(x) self-consistently determined from the non-coincident cumulants—by coupling it to an external field q(x) whose Boltzmann distribution is fixed by $F_0$.

We shall apply this result to give a new presentation of the known large deviation rate function in the classical SSEP. Recall that SSEP is a stochastic model suited for describing transport and density fluctuations in many particle systems out of equilibrium. Its rate function, denoted $I_\textrm{ssep}[n]$, governs the rare large density fluctuations in the sense that the probability that the SSEP density profile ${\mathfrak {n}}(x)$ approaches a given profile n(x) away from the mean, most probable profile is exponentially small :

$$\begin{aligned} {\mathbb {P}}\left[ {\mathfrak {n}}(\cdot ) \approx n(\cdot )\right] \asymp _{N\rightarrow \infty } e^{-N \,I_\textrm{ssep}[n] }, \end{aligned}$$

(8)

with N the number of sites. A more precise definition and description shall be given in Sect. 4.

The derivation of the new formula we shall give uses three ingredients : (i) the relation between SSEP and Q-SSEP [2]; (ii) the connections between the invariant measure of the quantum SSEP and free probability [7]; and (iii) the solution of the problem stated above.

Combining these first two ingredients leads to a representation of the generating function for the non-coincident cumulants of the density in the classical SSEP in terms of appropriate free cumulants. Namely, let $F^\textrm{ssep}_0[a]$ be the generating function of SSEP non-coincident cumulants, then

$$\begin{aligned} F^\textrm{ssep}_0[a] = \sum _{n\ge 1} \frac{(-1)^{n-1}}{n}R_n({\mathbb {I}}_{[a]}), \end{aligned}$$

where $R_n$ are the free cumulants of the function ${\mathbb {I}}_{[a]}(x)=\int _x^1\text {d}y\, a(y)$ viewed as a random variable on the interval [0, 1] equipped with the Lebesgue measure as probability measure.

Knowing the generating function of the non-coincident cumulants, we can then use the solution (6) of the problem stated above to write the large deviation rate function as the solution of the following extremization problem :

$$\begin{aligned} {I}_\textrm{ssep}[n]= & {} \max _{g(\cdot ), q(\cdot )} \left( \int _0^1\!\! \text {d}x \left[ n(x)\log \left( \frac{n(x)}{g(x)}\right) +(1-n(x))\log \left( \frac{1-n(x)}{1-g(x)}\right) +q(x)g(x)\right] -{F}^\textrm{ssep}_0[q]\right) .\nonumber \\ \end{aligned}$$

(9)

Comparing with the free formula (3), this formula has a mean field like self consistent flavor, as does the formula (6). It also shows similarities with the formula known in the SSEP literature [9, 10, 20] and we check in Sect. 4.5 that they of course coincide. Its derivation is, however, different, as it makes a detour through Q-SSEP and it reveals the hidden ingredients from free probability in the classical SSEP large deviation rate function.

Since the SSEP large deviation rate function has initially been derived using a matrix product ansatz for the SSEP stationary measure [9, 10, 13, 20], one may wonder if there is any connection between matrix product ansatz, or more generally tensor network techniques, and free probability. In view of the impact of tensor techniques in studies of quantum many-body systems, such connection, if it exists, would provide further evidence for the possible universal role of free probability tools in such systems [14, 23].

The rest of the paper is organized as follows. In Sect. 2, we show how to deal with cumulants of Bernoulli variables, using combinatorial techniques, and we derive the variational problem associated with the large N limit. Another approach to these results, using more standard Feynman diagram tools, is presented in Sect. 3. Finally, in Sect. 4, we make the connection with the Q-SSEP.

2 Bernoulli Partition Functions and Combinatorics

The purpose of this section is to give some combinatorial properties of cumulants, which will then be used to study the asymptotics of the free energy of a family of Bernoulli variables.

2.1 Partition Lattices and Möbius Functions

2.1.1 The Lattice of Partitions of a Finite Set

The set-partitions of $\{1,\ldots ,n\}$ (or, more generally, of a finite set S) form a lattice for the inverse refinement order, such that $\pi \le \gamma$ if $\pi$ is finer than $\gamma$. We denote by ${{\mathcal {P}}}_n$ (or ${{\mathcal {P}}}(S))$ this lattice. It has a maximal element $1_n$ (the partition with one part) and a minimal element $0_n$ (the partition with n parts). Every interval $[\pi _1,\pi _2]$ in this lattice is isomorphic, as a partially ordered set, to a product

$$\begin{aligned}{}[\pi _1,\pi _2]\sim \prod _{p}[0_{k_p},1_{k_p}] \end{aligned}$$

(10)

where the terms in the product are indexed by the parts p of $\pi _2$ and $k_p$ is the number of parts of $\pi _1$ which are subsets of p.

2.1.2 Lattice of Partitions of a Graph

Let G be a finite, simple and loopless¹ graph (all graphs considered below will satisfy these conditions) with set of vertices V and edges E and let ${{\mathcal {P}}}_G$ be the set of partitions of V into connected parts. Then, ${{\mathcal {P}}}_G\subset {{\mathcal {P}}}(V)$ with equality if and only if G is a complete graph. We endow this set with the inverse refinement order $<_G$.

For every partition $\pi$ of V there exists a maximal partition $\pi ^*\in {{\mathcal {P}}}_G$ such that $\pi ^*\le \pi$. The parts of this partition are the connected components of the parts of $\pi$. It follows that the partially ordered set ${{\mathcal {P}}}_G$ is a lattice with

$$\begin{aligned} \pi _1\vee _G\pi _2=\pi _1\vee \pi _2\qquad \pi _1\wedge _G\pi _2=(\pi _1\wedge \pi _2)^* \end{aligned}$$

Again there is a smallest element, $0_G$ and a maximal element $1_G$, whose parts are the connected components of G, moreover every interval $[\pi _1,\pi _2]$ is isomorphic to a lattice of the form ${{\mathcal {P}}}_{G'}$ for some graph $G'$.

Every partition $\pi \in {{\mathcal {P}}}_G$ defines a graph $G_\pi$ whose vertices are the parts of $\pi$ and two vertices are connected by an edge if and only if the union of the corresponding parts is connected in G. In terms of the graphs $G_\pi$, the covering relations for the order on ${{\mathcal {P}}}_G$ can be described as the contraction of an edge: $\pi _1<_G\pi _2$ is a covering relation if and only if $G_{\pi _2}$ can be obtained from $G_{\pi _1}$ by contracting some edge (and possibly removing spurious edges to keep the graph simple).

For example, here is ${{\mathcal {P}}}_G$ when G is a cycle of size 4. Each partition is denoted by its associated graph $G_\pi$.

2.1.3 Möbius Functions

Recall that, for a partially ordered set, its zeta function is the function

$$\begin{aligned} \begin{array}{rcl} \zeta (x,y)&{}=&{}1\quad \text {if } x\le y \\ &{}=&{} 0\quad \text {if not} \end{array} \end{aligned}$$

The Möbius function $\mu (x,y)$, defined for $x\le y$, satisfies, for all $x\le z$:

$$\begin{aligned} \sum _{y;x\le y\le z}\mu (x,y)\zeta (y,z)=\delta _{xz} \end{aligned}$$

The Möbius functions for the lattices ${\mathcal {P}}_n$ and, more generally, ${\mathcal {P}}_G$ play an important role in the following. The Möbius function on ${\mathcal {P}}_n$ is multiplicative namely if $[\pi _1,\pi _2]$ is as in (10) then

$$\begin{aligned} \mu (\pi _1,\pi _2)=\prod _p\mu (0_{k_p},1_{k_p}) \end{aligned}$$

and

$$\begin{aligned} \mu (0_k,1_k)=(-1)^{k-1}(k-1)! \end{aligned}$$

In order to compute the Möbius function on ${\mathcal {P}}_G$, we will need some facts about chromatic polynomials.

2.1.4 Chromatic Polynomials

A proper coloring of a finite graph G is a coloring of its vertices such that, for any edge, the adjacent vertices have different colors. The chromatic polynomial of G, denoted $\chi _G$, is the unique polynomial such that, for any integer $k\ge 1$, the number of proper colorings of G with at most k colors is equal to $\chi _G(k)$. If $\omega _r$ denotes the number of proper colorings of G which use exactly r colors, then one has

$$\begin{aligned} \chi _G(k)=\sum _r\omega _r{k\atopwithdelims ()r}. \end{aligned}$$

(11)

Since $\omega _r=0$ for $r>|V|$ this shows that $\chi _G$ is indeed a polynomial.

For example, the complete graph with n vertices has

$$\begin{aligned} \chi _{K_n}(z)=(z)_n:=z(z-1)(z-2)\ldots (z-n+1) \end{aligned}$$

while, if T is a tree with n vertices, then

$$\begin{aligned} \chi _{T}(z)=z(z-1)^{n-1}. \end{aligned}$$

We note the following properties of the chromatic polynomial: if G is the union of two disjoint graphs $G_1,G_2$, then

$$\begin{aligned} \chi _G(z)=\chi _{G_1}(z)\chi _{G_2}(z) \end{aligned}$$

(12)

whereas, if G is the join of $G_1,G_2$, namely $V=V_1\cup V_2$ and $V_1\cap V_2=\{v\}$ with no edge joining $V_1\setminus \{v\}$ to $V_2\setminus \{v\}$, then

$$\begin{aligned} \chi _G(z)=\frac{1}{z}\chi _{G_1}(z)\chi _{G_2}(z). \end{aligned}$$

(13)

The Möbius function of ${{\mathcal {P}}}_G$ has been computed by Rota [26], one has

$$\begin{aligned} \mu (0_G,1_G)=[z]\chi _G(z),\end{aligned}$$

(14)

the coefficient of z in the polynomial $\chi _G(z)$, moreover, if $\pi _1\le \pi _2$ in ${{\mathcal {P}}}_G$, then $[\pi _1,\pi _2]\sim {{\mathcal {P}}}_{G'}$ for some graph $G'$ and

$$\begin{aligned} \mu (\pi _1,\pi _2)=\mu (0_{G'},1_{G'}) \end{aligned}$$

Note that, by (11), one has

$$\begin{aligned}{}[z]\chi _G(z)=\sum _r\frac{(-1)^{r-1}}{r}\omega _r \end{aligned}$$

(15)

In the following we will use the notation $\mu (G)=\mu (0_G,1_G)$ when the context is clear.

The proof of (14) is based on inclusion–exclusion. The number of all colorings of G using at most k colors is $k^{|V|}$, moreover any such coloring determines a partition $\pi \in {{\mathcal {P}}}_G$ into connected unicolor components, so that the graph $G_\pi$ is properly colored. It follows that

$$\begin{aligned} k^{|V|}=\sum _{\pi \in {{\mathcal {P}}}_G}\chi _{G_\pi }(k) \end{aligned}$$

and formula (14) is obtained by Möbius inversion, see [26] for details.

2.2 Moments and Cumulants

Let ${{\mathcal {A}}}$ be a complex algebra with unit and $\varphi :{{\mathcal {A}}}\rightarrow \textbf{C}$ a linear form such that $\varphi (1)=1$. For most applications below ${{\mathcal {A}}}$ will be an algebra of complex random variables defined over some probability space, in particular it will be commutative, but it is not more difficult to consider here the general case of an arbitrary algebra over the complex numbers.

The cumulants are a sequence of n-multilinear forms $K_n, n=1,2,\ldots$ on ${{\mathcal {A}}}$, implicitly defined by

$$\begin{aligned} \varphi (a_1\ldots a_n)=\sum _{\pi \in {{\mathcal {P}}}_n} K_\pi (a_1,\ldots ,a_n) \end{aligned}$$

(16)

with

$$\begin{aligned} K_\pi (a_1,\ldots ,a_n)=\prod _{p\in \pi }K_{|p|}(a_{i_1},\ldots ,a_{i_{|p|}}) \end{aligned}$$

(17)

the product being over the parts of $\pi$ with $p=\{i_1,\ldots ,i_{|p|}\}$ and $i_1<i_2<\ldots <i_{|p|}$. This formula can be inverted to express the cumulants in terms of the “moments,” i.e., $\varphi$ evaluated on products. For example,

$$\begin{aligned} \varphi (a_1)= & {} K_1(a_1) \\ \varphi (a_1a_2)= & {} K_2(a_1,a_2)+K_1(a_1)K_1(a_2) \end{aligned}$$

gives

$$\begin{aligned} K_2(a_1,a_2)=\varphi (a_1a_2)-\varphi (a_1)\varphi (a_2) \end{aligned}$$

while

$$\begin{aligned} \begin{array}{rcl}\varphi (a_1a_2a_3)&{}=&{}K_3(a_1,a_2,a_3)+ K_1(a_1)K_2(a_2,a_3)+ K_2(a_1,a_3)K_1(a_2)\\ {} &{}&{} +K_2(a_1,a_2)K_1(a_3)+ K_1(a_1)K_1(a_2)K_1(a_3)\end{array} \end{aligned}$$

gives

$$\begin{aligned} \begin{array}{rcl} K_3(a_1,a_2,a_3)&{}=&{}\varphi (a_1a_2a_3)-\varphi (a_1a_2)\varphi (a_3)-\varphi (a_1a_3)\varphi (a_2)\\ {} &{}&{} -\varphi (a_1)\varphi (a_2a_3) +2\varphi (a_1)\varphi (a_2)\varphi (a_3)\end{array} \end{aligned}$$

In the general case there is an expression using the Möbius function on ${{\mathcal {P}}}_n$:

$$\begin{aligned} K_n(a_1,\ldots ,a_n)=\sum _{\pi \in {{\mathcal {P}}}_n} \varphi _\pi (a_1,\ldots , a_n)\mu (\pi ,1_n) \end{aligned}$$

In the case where ${{\mathcal {A}}}$ is commutative the cumulants are symmetric multilinear forms and their generating function is

$$\begin{aligned} \log \varphi [e^{\sum _{i=1}^N\lambda _ia_i}]=\sum _{n=1}^\infty \frac{1}{n!}\sum _{I:i_1+\ldots +i_N=n}\lambda _1^{i_1}\ldots \lambda _N^{i_N}K_n(a_I) \end{aligned}$$

(18)

where one sums over all sequences $a_I=(a_{j_1},\ldots , a_{j_n})$ with $i_k$ occurrences of $a_k$. Each such sequence determines a partition of $\{1,\ldots ,n\}$ into parts corresponding to the value of the indices. One can thus rewrite (18) as

$$\begin{aligned} \log \varphi [e^{\sum _{i=1}^N\lambda _ia_i}]=\sum _{n=1}^\infty \frac{1}{n!}\sum _{\Gamma \in {\mathcal LP}_n}\lambda ^\Gamma K_n(a_\Gamma ) \end{aligned}$$

(19)

where the sum is over labeled partitions $\Gamma$ of $\{1,\ldots ,n\}$ into at most N parts, where each part $\gamma$ of $\Gamma$ has a label $\nu (\gamma )$ in $\{1,2,\ldots ,N\}$ (the parts having distinct labels) and $\lambda ^\Gamma =\prod _{\gamma \in \Gamma }\lambda _{\nu (\gamma )}^{|\gamma |}$.

2.2.1 Cumulants with Products as Entries

Let $\Gamma :=\gamma _1\cup \ldots \cup \gamma _k$ be a partition of $\{1,\ldots ,n\}$ into intervals, i.e., each $\gamma _l$ is of the form $\{j_l+1,j_l+2,\ldots ,j_{l+1}\}$ with $0=j_1<j_2<\ldots <j_{k+1}=n$.

Let us define

$$\begin{aligned} K_n^{\Gamma }(a_1,\ldots ,a_n)=K_k(A_1,\ldots ,A_k) \end{aligned}$$

where $A_l=a_{j_l+1}a_{j_l+2}\ldots a_{j_{l+1}}$, the product of the $a_i$ with indices $i\in \gamma _l$ and, more generally,

$$\begin{aligned} K^\Gamma _\pi (a_1,\ldots ,a_n)=\prod _{p\in \pi }K_{|p|}^{\Gamma _{|p}}(a_{i_1},\ldots ,a_{i_{|p|}}) \end{aligned}$$

(20)

Here, $\Gamma _{|p}$ is the partition of $p\in \pi$ induced by $\Gamma$. Observe that one has also

$$\begin{aligned} K^\Gamma _\pi (a_1,\ldots ,a_n)=K_\pi ^{\Gamma \wedge \pi }(a_1,\ldots ,a_n). \end{aligned}$$

(21)

One has

$$\begin{aligned} \begin{array}{rcl} K^{0_n}_\pi (a_1,\ldots ,a_n)&{}=&{}K_\pi (a_1,\ldots ,a_n)\\ K^{1_n}_\pi (a_1,\ldots ,a_n)&{}=&{}\varphi _\pi (a_1\ldots a_n) \end{array} \end{aligned}$$

so that the $K^\Gamma _\pi$, for $0_n\le \Gamma \le 1_n$, interpolate between cumulants and moments. The following formula, attributed to Leonov and Shiryaev [19], expresses the $K^\Gamma$ in terms of ordinary cumulants.

Theorem 1

$$\begin{aligned} K^{\Gamma }_\xi (a_1,\ldots ,a_n)=\sum _{\pi :\pi \vee \Gamma =\xi }K_\pi (a_1,\ldots ,a_n),\qquad \text {for}\quad \xi \ge \Gamma . \end{aligned}$$

(22)

In particular

$$\begin{aligned} K_n^{\Gamma }(a_1,\ldots ,a_n)=\sum _{\pi :\pi \vee \Gamma =1_n}K_\pi (a_1,\ldots ,a_n). \end{aligned}$$

(23)

Proof

This follows easily by comparing the two formulas:

$$\begin{aligned} \begin{array}{rcl} \varphi (a_1\ldots a_n)&{}=&{}\sum \limits _{\pi }K_\pi (a_1,\ldots ,a_n)=\sum \limits _{\xi \ge \Gamma }\left( \sum _{\pi :\pi \vee \Gamma =\xi }K_\pi (a_1,\ldots ,a_n)\right) \\ \varphi (a_1\ldots a_n)&{}=&{}\varphi (A_1\ldots A_n)=\sum \limits _{\xi \ge \Gamma }K_\xi ^\Gamma (a_1,\ldots ,a_n) \end{array} \end{aligned}$$

$\square$

When $\Gamma =0_n$ the formula (23) is trivially true and when $\Gamma =1_n$ it is the moments-cumulants formula (16). Also if $\xi \ngeq \Gamma$ one can use (21) to get

$$\begin{aligned} K^{\Gamma }_\xi (a_1,\ldots ,a_n)=K^{\Gamma \wedge \xi }_\xi (a_1,\ldots ,a_n)=\sum _{\pi :\pi \vee (\Gamma \wedge \xi )=\xi }K_\pi (a_1,\ldots ,a_n) \end{aligned}$$

(24)

In the general case, the formula (23) can be inverted. For this, we introduce a graph $G_{\pi ,\Gamma }$, whose vertices are the parts of $\pi$, and there is an edge between p and q if there exists a part $\gamma$ of $\Gamma$ such that $p\cap \gamma \ne \emptyset$ and $q\cap \gamma \ne \emptyset$. One has then

Theorem 2

$$\begin{aligned} K_n(a_1,\ldots ,a_n)=\sum _{\pi :\pi \vee \Gamma =1_n}K_\pi ^\Gamma (a_1,\ldots ,a_n)\mu (G_{\pi ,\Gamma }) \end{aligned}$$

(25)

Proof

This formula can be verified by plugging it into the right-hand side of (22) and checking that it reduces to $K_n^{\Gamma }(a_1,\ldots ,a_n)=K_n^{\Gamma }(a_1,\ldots ,a_n)$ after using the properties of the Möbius function.

Introducing the unknown function $\mu _\Gamma (\pi )$ such that

$$\begin{aligned} K_n(a_1,\ldots ,a_n)=\sum _{\pi :\pi \vee \Gamma =1_n}K_\pi ^\Gamma (a_1,\ldots ,a_n)\mu _\Gamma (\pi ) \end{aligned}$$

and using (24) one has

$$\begin{aligned} \begin{array}{rcl} K_n(a_1,\ldots ,a_n)&{}=&{}\sum \limits _{\pi :\pi \vee \Gamma =1_n}K_\pi ^\Gamma (a_1,\ldots ,a_n)\mu _\Gamma (\pi )\\ &{}=&{}\sum \limits _{\pi :\pi \vee \Gamma =1_n}\sum \limits _{\xi :(\pi \wedge \Gamma )\vee \xi =\pi }K_\xi (a_1,\ldots ,a_n)\mu _\Gamma (\pi )\\ &{}=&{}\sum \limits _{\xi :\xi \vee \Gamma =1_n}K_\xi (a_1,\ldots ,a_n)\left( \sum \limits _{\pi :(\pi \wedge \Gamma )\vee \xi =\pi }\mu _\Gamma (\pi )\right) \\ \end{array} \end{aligned}$$

so that $\mu _\Gamma$ has to satisfy

$$\begin{aligned} \sum \nolimits _{\pi :(\pi \wedge \Gamma )\vee \xi =\pi }\mu _\Gamma (\pi )=1,\quad \text {if}\quad \xi =1_n, \quad =0\quad \text {if not.} \end{aligned}$$

(26)

Define an order relation on the set of partitions $\pi$ such that $\pi \vee \Gamma =1_n$ by requiring

$$\begin{aligned} \pi _1\le _\Gamma \pi _2\quad \text { if and only if}\quad \pi _1\le \pi _2\quad \text {and}\quad \pi _1\vee (\pi _2\wedge \Gamma )=\pi _2 \end{aligned}$$

Indeed, if $\pi _1\le _{\Gamma }\pi _2$ and $\pi _2\le _{\Gamma }\pi _3$, then $\pi _1\vee (\pi _3\wedge \Gamma )\ge \pi _1\vee (\pi _2\wedge \Gamma )=\pi _2$; therefore, $\pi _1\vee (\pi _3\wedge \Gamma )\ge \pi _2$ and $\pi _1\vee (\pi _3\wedge \Gamma )\ge \pi _3\wedge \Gamma$ so that, finally $\pi _1\vee (\pi _3\wedge \Gamma )\ge \pi _2\vee (\pi _3\wedge \Gamma )=\pi _3$. This relation is transitive as claimed. Taking $\mu _\Gamma (\pi )=\mu ([\pi ,1_n])$ where $\mu$ is the Möbius function for this order yields (26). It is easy to check that the interval $[\pi ,1_n]$ for this order is isomorphic à ${{\mathcal {P}}}_{G_{\pi ,\Gamma }}$. One has thus $\mu _\Gamma (\pi )=[z]\chi _{G_{\pi ,\Gamma }}=\mu (G_{\pi ,\Gamma })$.

$\square$

In the commutative case, one can define the cumulants $K^{\Gamma }$ for any partition $\Gamma$, not just interval partitions, since the cumulants are symmetric and (22), (23), (25) still hold.

2.3 Non-crossing Partitions and Cumulants

In this section, we define non-crossing cumulants which will be used later in the asymptotic analysis of the Q-SSEP. Many information about the combinatorics of the non-crossing cumulants may be found in the book by Nica and Speicher [22].

A partition of $\{1,\ldots ,n\}$ has a crossing if there exists two parts of the partition and $i<j<k<l$ such that i, k belong to the first part and j, l to the second part. Partitions without crossing are called non-crossing. The set of non-crossing partitions of $\{1,\ldots ,n\}$, denoted NC(n), is a lattice under the inverse refinement order, and each interval $[\pi _1,\pi _2]$ is isomorphic, as a partially ordered set, to a product $\prod _iNC(k_i)$ for some integers $k_i$. The Möbius function is again multiplicative and one has

$$\begin{aligned} \mu _{NC(n)}(0_n,1_n)=(-1)^{n-1}\text {Cat}_{n-1} \end{aligned}$$

where $\text {Cat}_n=\frac{1}{n+1}{2n\atopwithdelims ()n}$ is a Catalan number.

Non-crossing cumulants $R_n$ are defined similarly as the cumulants $K_n$ using an implicit formula:

$$\begin{aligned} \varphi (a_1\ldots a_n)=\sum _{\pi \in NC(n)} R_\pi (a_1,\ldots ,a_n) \end{aligned}$$

(27)

which can be inverted as

$$\begin{aligned} R_n(a_1,\ldots ,a_n)=\sum _{\pi \in NC(n)} \varphi _\pi (a_1,\ldots , a_n)\mu _{NC(n)}(\pi ,1_n) \end{aligned}$$

Every partition $\pi \in {{\mathcal {P}}}_n$ has a least non-crossing majorant ${\hat{\pi }}$. Using this one can write

$$\begin{aligned} \begin{array}{rcl} \varphi (a_1\ldots a_n)&{}=&{}\sum \limits _{\pi \in {{\mathcal {P}}}_n }K_\pi (a_1,\ldots ,a_n)=\sum \limits _{\xi \in NC(n)}\left( \sum \limits _{{\hat{\pi }}=\xi }K_\pi (a_1,\ldots ,a_n)\right) \\ \varphi (a_1\ldots a_n)&{}=&{}\sum \limits _{\xi \in NC(n)}R_\xi (a_1,\ldots ,a_n) \end{array} \end{aligned}$$

from which one can easily deduce that

$$\begin{aligned} R_\xi (a_1,\ldots ,a_n)=\sum _{\pi :{\hat{\pi }}=\xi }K_\pi (a_1,\ldots ,a_n). \end{aligned}$$

In particular, the relation

$$\begin{aligned} R_n(a_1,\ldots ,a_n)=\sum _{\pi :{\hat{\pi }}=1_n}K_\pi (a_1,\ldots ,a_n) \end{aligned}$$

(28)

expresses non-crossing cumulants in terms of cumulants. The formula can be reversed using again the Möbius function of a certain lattice ${{\mathcal {P}}}_G$. For this, define the crossing graph $G^c_\pi$ of a partition $\pi$ as the graph whose vertices are the parts of $\pi$ and two parts of $\pi$ are connected if they contain a crossing. Using this graph one has, by Möbius inversion,

Proposition 3

$$\begin{aligned} K_n(a_1,\ldots ,a_n)=\sum _{\pi :{\hat{\pi }}=\xi }R_\pi (a_1,\ldots ,a_n)\mu (G^c_\pi ). \end{aligned}$$

(29)

An equivalent formula was first derived in [15] by different means, see also [1].

There is also, for non-crossing cumulants, an analog of the formula (23), due to Krawczyk and Speicher [17].

2.4 Cumulants of Bernoulli Variables

2.4.1 Non-coincident Cumulants

Let $b_i; i=1,2,\ldots ,N$ be a sequence of (commuting) Bernoulli random variables, taking values in $\{0,1\}$. They satisfy $b_i^r=b_i$ for all $r\ge 1$; therefore, all the information about the joint distribution of the $b_i$ is contained in the $2^N-1$ “non-coincident moments,” i.e., the quantities $E[b_{i_1}b_{i_2}\ldots b_{i_k}]$ where $1\le i_1<i_2<\ldots <i_k\le N$ (here E denotes the expectation), or in the $2^N-1$ “non-coincident cumulants” $K_k(b_{i_1},b_{i_2},\ldots ,b_{i_k})$. It is therefore of interest to express an arbitrary cumulant $K_n(b_{j_1},\ldots ,b_{j_n})$, for a sequence of indices $1\le j_k\le N$, in terms of these non-coincident cumulants.

Let $\Gamma$ be the partition of $\{1,\ldots ,n\}$ such that k and l are in the same part of $\Gamma$ if and only if $i_k=i_l$. Using the fact that $(b_i)^r=b_i$ for any $r\ge 1$ we see that the $\Gamma$-cumulants defined by (20) and the formula (25) express any cumulant as a polynomial in the non-coincident cumulants.

$$\begin{aligned} K_n(b_{i_1},\ldots ,b_{i_n})=\sum _{\pi :\pi \vee \Gamma =1_n}K_\pi ^\Gamma (b_{i_1},\ldots ,b_{i_n})\mu (G_{\pi ,\Gamma }) \end{aligned}$$

(30)

2.4.2 Free Energy

Recall the generating function of the cumulants (19)

$$\begin{aligned} \log E[e^{\sum _{i=1}^Nh_ib_i}]=\sum _{n=1}^\infty \frac{1}{n!}\sum _{\Gamma }h^\Gamma K_n(b_\Gamma ) \end{aligned}$$

(31)

One can use (30) in each term of this sum to obtain a sum over pairs $(\pi ,\Gamma )$:

$$\begin{aligned} \log E[e^{\sum _{i=1}^Nh_ib_i}]=\sum _n\frac{1}{n!}\sum _\Gamma h^\Gamma \left( \sum _{\pi :\pi \vee \Gamma =1_n} K_\pi ^\Gamma (b_\Gamma )\mu (G_{\pi ,\Gamma })\right) . \end{aligned}$$

(32)

Let us introduce a labeled bipartite graph $\Delta _{\pi ,\Gamma }$ with the parts of $\Gamma$ as set of white vertices and the parts of $\pi$ as set of black vertices. The white vertices, corresponding to the parts of $\Gamma$, are labeled by $1,2,\ldots$ (the indices of the Bernoulli variables) each index appearing at most once. There is an edge between a part of $\Gamma$ and a part of $\pi$ if they have a non-empty intersection. The condition $\pi \vee \Gamma =1_n$ ensures that this graph is connected. Observe that one can associate to every edge of the graph a subset of $\{1,\ldots ,n\}$ by taking the intersection of the part of $\pi$ corresponding to its black extremity and the part of $\Gamma$ corresponding to its white extremity. These sets form a partition of $\{1,\ldots ,n\}$, indexed by the edges of the graph, and one can reconstruct the partitions $\pi$ and $\Gamma$ from this edge-indexed partition by taking the union over edges adjacent to a white vertex to get the parts of $\Gamma$ or to a black vertex, to get the parts of $\pi$.

The graph $G_{\pi ,\Gamma }$ is obtained from the bipartite graph $\Delta _{\pi ,\Gamma }$ by keeping the black vertices and putting an edge between two such vertices if they have at least one white neighbor in common.

The factor associated with the pair $\pi ,\Gamma$ can then be written as

$$\begin{aligned} \mu (G_{\pi ,\Gamma })\frac{h^\Gamma }{n!}\prod _{\bullet }K(b_\bullet ) \end{aligned}$$

(33)

where the product is over the black vertices $\bullet$ of $\Delta _{\pi ,\Gamma }$ and, for each such vertex, the factor

$$\begin{aligned} K(b_\bullet )= K_k(b_{u_1},b_ {u_2},\ldots ,b_{u_k}) \end{aligned}$$

the indices $u_1,u_2,\ldots , u_k$ being those of the white neighbors of $\bullet$ in $\Delta _{\pi ,\Gamma }$.

As an example, here is the graph $\Delta _{\pi ,\Gamma }$ associated with the partitions

$$\begin{aligned} \pi =\{1,2,5,8,14\}\cup \{3,6,10,12\}\cup \{4,7,9,11,13\} \end{aligned}$$

and

$$\begin{aligned} \Gamma =\{1,5,13,14\}\cup \{2,6,8,9,11\}\cup \{3,4,7\}\cup \{10,12\} \end{aligned}$$

where we show, near each edge, the associated set:

Let H be a bipartite connected graph (with at least two vertices), then the pairs $(\pi ,\Gamma )$ such that H is the underlying unlabeled graph of $\Delta _{\pi ,\Gamma }$ can be obtained by

• 1. Labeling the white vertices of H with distinct labels in $\{1,2,\ldots ,N\}$ (we call Lab(H) the set of such labelings).

• 2. Choosing a partition of $\{1,\ldots ,n\}$ indexed by the edges of H.

Denote $H^\bullet$ the graph whose vertices are the black vertices of H and with edges between vertices sharing a white neighbor in H. Using the fact that we are summing over partitions indexed by edges of H, which are counted by multinomial coefficients, we see that the sum of all contributions (33) corresponding to H is

$$\begin{aligned} \frac{\mu (H^\bullet )}{|\text {Aut}\, H|}\sum _{{{\mathcal {L}}}\in Lab(H)}w({{\mathcal {L}}}) \end{aligned}$$

(34)

where ${{\mathcal {L}}}$ runs over all labelings of the white vertices of H by distinct indices and

$$\begin{aligned} w({{\mathcal {L}}})=\prod _{\bullet }K(b_\bullet )\prod _{\hbox { edges of}\ H}e_i \end{aligned}$$

(35)

(for any edge e of H one denotes $e_i:=e^{h_i}-1$, where i is the index of the white vertex adjacent to the edge e). The term $\frac{1}{|\text {Aut}\, H|}$, as usual, is here to avoid overcountings due to symmetries. The automorphism group is that of H considered as a bipartite graph, i.e., automorphisms should send black vertices to black vertices and white vertices to white vertices. We can thus rewrite (32) as

Proposition 4

$$\begin{aligned} W[h]=\log E\big [e^{\sum _{i=1}^Nh_ib_i}\big ]=\sum _H\frac{\mu (H^\bullet )}{|\text {Aut}\, H|}\sum _{{{\mathcal {L}}}\in Lab(H)}w({{\mathcal {L}}}), \end{aligned}$$

(36)

where the sum is over connected bipartite graphs and the weight $w({{\mathcal {L}}})$ as in Eq. (35).

Here, the graph H corresponding to the pair $\pi ,\Gamma$ above, with a labeling by 1, 2, 5, 6.

The graph $H^\bullet$ is a complete graph with three vertices so that $\mu (H^\bullet )=2$ and there are no nontrivial automorphisms moreover the weight of the labeling is

$$\begin{aligned} w({{\mathcal {L}}})=e_1^2e_2^3e_5e_6^2K_3(b_1,b_2,b_6)K_2(b_2,b_6)K_3(b_1,b_2,b_5) \end{aligned}$$

2.4.3 Another Proof of Formula (36)

We sketch another derivation of (36), which does not rely on the theory of cumulants of products. One has

$$\begin{aligned} e^{\sum _{i=1}^Nh_ib_i}=\prod _i(1+e_ib_i)=1+\sum _{I\subset \{1,\ldots ,N\};I\ne \emptyset }e_Ib_I \end{aligned}$$

where $e_I$ is the product $\prod _{i\in I}e_i$. Using the moment-cumulant formula, we get

$$\begin{aligned} E[e^{\sum _{i=1}^Nh_ib_i}]=1+\sum _{I\ne \emptyset }e_I\sum _{\pi \in {{\mathcal {P}}}(I)}K_\pi (b_I) \end{aligned}$$

(37)

and taking the logarithm

$$\begin{aligned} W[h]=\sum _{r=1}^\infty \frac{(-1)^{r+1}}{r}\left( \sum _{I\ne \emptyset }e_I\sum _{\pi \in {{\mathcal {P}}}(I)}K_\pi (b_I)\right) ^r \end{aligned}$$

One has

$$\begin{aligned} \left( \sum _{I\ne \emptyset }e_I\sum _{\pi \in {{\mathcal {P}}}(I)}K_\pi (b_I)\right) ^r=\sum _{I_1,\ldots ,I_r,\pi _1\ldots ,\pi _r}\prod _{k=1}^rK_{\pi _k}(b_{I_k})e_{I_k} \end{aligned}$$

where we sum over $I_1\ldots , I_r$, non-empty subsets of [1, N] and $\pi _k$ partition of $I_k$.

We now introduce a bipartite graph with white vertices labeled by the $i\in \cup _k I_k$ and black vertices corresponding to the parts of the partitions $\pi _k$. There is an edge between a white and a black vertex if the index of the white vertex is in the part corresponding to the black vertex. This bipartite graph induces a graph structure on the black vertices: two vertices share an edge if they have a common white neighbor. For each such graph, we have to sum over all proper colorings of the black vertices using exactly r colors. Using relation (11), we identify the combinatorial term associated with a graph to the z coefficient in the chromatic polynomial of the black graph. By (12) this coefficient is zero is the graph is not connected so that the sum can be taken over connected graphs We leave details to the reader and give an example: it is easy to see that the monomial

$$\begin{aligned} e_1^3e_2^3K_1(b_1)^2K_1(b_2)^2K_2(b_1,b_2) \end{aligned}$$

is obtained from only one graph H, the one depicted below.

(38)

The graph $H^\bullet$ is the joining of two complete graphs therefore $\mu (H^\bullet )=4$ while $|\text {Aut}(H)|=8$; moreover, there are two labelings of the white vertices by 1, 2; therefore, the sum of coefficients of this graph is 1, which should be the coefficient of the monomial.

On the other hand, one can obtain the coefficient of this monomial by expanding the expression $\frac{1}{3}w^3-\frac{1}{4}w^4+\frac{1}{5}w^5$ (other powers of w do not contribute) where

$$\begin{aligned} w=e_1K_1(b_1)+e_2K_1(b_2)+e_1e_2K_2(b_1,b_2)+e_1e_2K_1(b_1)K_1(b_2) \end{aligned}$$

Using multinomial coefficients we find

$$\begin{aligned} \frac{1}{3}\frac{3!}{2!1!}-\frac{1}{4}\frac{4!}{1!1!1!}+\frac{1}{5}\frac{5!}{1!2!2!}=1, \end{aligned}$$

so that the weight of this graph is effectively 1.

2.5 Asymptotic Behavior and Legendre Transform

2.5.1 Reduction to Trees

We suppose now that, as $N\rightarrow \infty$, the non-coincident cumulants have a specific asymptotic behavior: there exists some compact space $\Sigma$, some functions $\rho _N:[1,N]\rightarrow \Sigma$ and continuous functions $\psi _n$ on $\Sigma ^n$ such that as $N\rightarrow \infty$

$$\begin{aligned} K_n(b_{i_1},\ldots ,b_{i_n})\sim N^{1-n}\psi _n(\rho _N(i_1),\ldots ,\rho _N(i_n)) \end{aligned}$$

(39)

Moreover the measures $\frac{1}{N}\sum _i\delta _{\rho _N(i)}$ converge to some diffuse measure ds on $\Sigma$. We are mainly interested in the case where $\rho _N(i)=i/N$ and $\Sigma =[0,1]$ but the analysis works in greater generality and can be adapted to deal with other topologies, e.g., lattices in higher dimension.

We will study the asymptotic behavior of the free energy and for this we assume that the $h_i$ converge also to some bounded function h(s) on $\Sigma$. We can then estimate the contribution of a graph H in (36). Indeed, the number of labelings is

$$\begin{aligned} (N)_{\sharp \text {white vertices}}\sim N^{\sharp \text {white vertices}} \end{aligned}$$

while, by (39), the contribution of the product of cumulants is of the order

$$\begin{aligned} O(N^{\sharp \text {black vertices}-\sharp \text {edges}}) \end{aligned}$$

It follows that the contribution coming from trees, for which

$$\begin{aligned} \sharp \text {white vertices}+\sharp \text {black vertices}-\sharp \text {edges}=1 \end{aligned}$$

is of the order O(N), while the contribution of other graphs is of lower order in N. For H a tree, the combinatorial factor is $\mu (H^\bullet )=\prod _i (-1)^{k_i-1}(k_i-1)!$ the product being over white vertices and $k_i$ being the number of black neighbors of the white vertex indexed by i. This follows from the computation of the chromatic polynomial for the complete graph and the formula for the joining of two graphs given by (13). In this case, we can thus rewrite

$$\begin{aligned} \frac{\mu (H^\bullet )}{|\text {Aut}\,H|}w({{\mathcal {L}}})=\frac{1}{|\text {Aut}(H)|}\prod _\bullet K_\bullet (b_\bullet )\prod _{\circ }(-1)^{k_\circ -1}(k_\circ -1)!\prod _ee_i \end{aligned}$$

where the product $\prod _\circ$ is over white vertices and $k_\circ$ denotes the number of neighbors of the white vertex $\circ$.

2.5.2 Gradient of the Free Energy

Let us now compute $e_i\frac{\partial W}{\partial e_i}$ in the large N limit. Since, in the weight $w({{\mathcal {L}}})$, there is a factor $e_i$ for each edge adjacent to a white vertex labeled i, one sees that this derivative is given by a sum over pairs T, e of a tree T and an edge e of T

$$\begin{aligned} e_i\frac{\partial W}{\partial e_i} \sim \sum _T\frac{\mu (T^\bullet )}{|\text {Aut}\,T|}\sum _{e\,\text {edge of}\, T}\sum _{{{\mathcal {L}}};e_\circ \sim i}w({{\mathcal {L}}}) \end{aligned}$$

(40)

where we sum over all labelings ${\mathcal {L}}$ such that the white vertex of e is labeled by i. Cutting the edge e splits the tree into two rooted trees, $T_\bullet$ and $T_\circ$, one of them containing the black vertex of e as a root and the other the white vertex, with labelings ${{\mathcal {L}}}_\bullet ,{{\mathcal {L}}}_\circ$. The automorphism subgroup fixing e is the product of the automorphism groups of these two rooted trees (that is, the automorphisms fixing the roots): $\text {Aut}_e(T)\sim {\text {Aut}(T_\bullet })\times \text {Aut}(T_\circ )$, while the term

$$\begin{aligned} \prod _\bullet K_\bullet (b_\bullet )\prod _{\circ }(-1)^{k_\circ -1}(k_\circ -1)!e_i\prod _{e'\ne e}e_j \end{aligned}$$

splits into a product over the two trees. One can sum over all edges in the orbit of e by $\text {Aut}(T)$ (whose size is $|\text {Aut}(T)|/|\text {Aut}_e(T))$ and get a sum

$$\begin{aligned} e_i\frac{\partial W}{\partial e_i} \sim \sum _T\sum _{e\in E(T)/\text {Aut}(T)}\frac{1}{|\text {Aut}(T_\bullet )||\text {Aut}(T_\circ )|}\sum _{{{\mathcal {L}}}; e_\circ \sim i}z_i^\bullet ({{\mathcal {L}}}_\bullet )z_i^\circ ({{\mathcal {L}}}_\circ ) \end{aligned}$$

(41)

In this sum the weights $z_i^\bullet ,z_i^\circ$ are computed on labeled bipartite trees with a black (resp. white) root and one has

$$\begin{aligned} z_i^\bullet ({{\mathcal {L}}}_\bullet )=e_iK_{root}(b_i,\ldots )\prod _{\bullet \ne root} K_\bullet (b_\bullet )\prod _{\circ }(-1)^{k_\circ -1}(k_\circ -1)!\prod _{e'}e_j \end{aligned}$$

for a tree with a black root, where $K_{root}(b_i,\ldots )$ is the non-coincident cumulant evaluated on $b_i$ and the neighbors of the black root. Similarly,

$$\begin{aligned} z_i^\circ ({{\mathcal {L}}}_\circ )=(-1)^{k_{root}}k_{root}!\prod _{\bullet } K_\bullet (b_\bullet )\prod _{\circ \ne root}(-1)^{k_\circ -1}(k_\circ -1)!\prod _{e'}e_j \end{aligned}$$

for a tree with a white root. Let us introduce the functions

$$\begin{aligned} q_i=\sum _{T_{\circ }}\frac{1}{|\text {Aut}(T_\circ )|}\sum _{{{\mathcal {L}}}_\circ }z_i^\circ ({{\mathcal {L}}}_\circ ) \end{aligned}$$

(42)

$$\begin{aligned} g_i=\sum _{T_{\bullet }}\frac{1}{|\text {Aut}(T_\bullet )|}\sum _{{{\mathcal {L}}}_\bullet } z_i^\bullet ({{\mathcal {L}}}_\bullet ) \end{aligned}$$

(43)

If we compare the expression on the rhs of (41) with the product $g_iq_i$, we see that they coincide up to possibly some repetitions in the labelings in the expansion of qg (since we consider the product of labelings of $T_\bullet$ and $T_\circ$); however, the number of terms with repetition in the labelings is of smaller order in N; therefore, as $N\rightarrow \infty$, one has

$$\begin{aligned} e_i\frac{\partial W}{\partial e_i}\sim g_i q_i. \end{aligned}$$

One can depict the trees and the weights involved in the definition of the functions g and q as follows:

2.5.3 A Variational Principle

Reasoning as in (2.5.2) by cutting the tree in either (42) or (43) at its root to form a forest, we find the following relations in the continuous limit between $q(s),g(s),e(s):=e^{h(s)}-1$:

$$\begin{aligned} \begin{array}{rcl} e(s)\frac{\partial W}{\partial e(s)}&{}=&{}g(s)q(s)\\ q(s)&{}=&{}\frac{e(s)}{1+e(s)g(s)}\\ g(s)&{}=&{}\frac{\delta }{\delta q(s)}F_0(q) \end{array} \end{aligned}$$

where $F_0(q)$ is the large N limit of the generating functional of the cumulants:

$$\begin{aligned} F_0(q)=\sum _n\frac{1}{n!}\int _{\Sigma ^n} q(s_1)q(s_2)\ldots q(s_n)\psi _n(s_1,\ldots ,s_n)ds_1\ldots ds_n \end{aligned}$$

It follows that

Proposition 5

In the scaling limit, the free energy is obtained by solving the following variational problem

$$\begin{aligned} \lim _{N\rightarrow \infty }\frac{1}{N}W=\max _{g,q}\left[ \int [\log (1+e(s)g(s))-q(s)g(s)] ds+F_0(q)\right] \end{aligned}$$

(44)

We shall use this variational formula in the case of SSEP in Sect. 4.

3 Bernoulli Partition Functions and Feynman Graphs

In this section, we employ standard field theory techniques (mainly the semiclassical expansion and Feynman graphs) to encode the combinatorics of Bernoulli cumulants.

3.1 Integral Representation of $\log Z$

We start with another description of $\log Z$ in terms of a graphical expansion. The starting point is a somehow tautological representation of Z as a formal Gaussian integral (see subsection A.2 and the following for some background if needed).

The basic observation is that, J being an arbitrary index set, $({{\overline{\lambda }}}_i)_{i\in J}$ and $(\lambda _I)_{I {\mathop {\subset }\limits ^{\circ }} J}$ being formal variables (the notation $I {\mathop {\subset }\limits ^{\circ }} J$ means that I is a finite, non-empty subset of J):

$$\begin{aligned}{} & {} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi } \exp (- z_i {{\overline{z}}}_i) (1+{{\overline{\lambda }}}_iz_i)\right) \\{} & {} \quad \exp \left( \sum _{I {\mathop {\subset }\limits ^{\circ }} J} \lambda _I {{\overline{z}}}_I\right) = 1+\sum _{I {\mathop {\subset }\limits ^{\circ }} J} {{\overline{\lambda }}}_I \left( \sum _{\pi \in {{\mathcal {P}}}(I)} \lambda _{\pi }\right) ,\end{aligned}$$

where ${{\overline{\lambda }}}_I:=\prod _{i\in I} {{\overline{\lambda }}}_i$ (and analogously ${{\overline{z}}}_I:=\prod _{i\in I} {{\overline{z}}}_i$), ${{\mathcal {P}}}(I)$ is the set of partitions of I and, for $I\ne \emptyset$ and $\pi :I=\sqcup _{\alpha } I_{\alpha } \in {{\mathcal {P}}}(I)$, $\lambda _{\pi }:= \prod _{\alpha } \lambda _{I_{\alpha }}$. This formula is checked by expanding

$$\begin{aligned} \prod _{i\in J} (1+{{\overline{\lambda }}}_iz_i)=1+\sum _{I {\mathop {\subset }\limits ^{\circ }} J} {{\overline{\lambda }}}_Iz_I.\end{aligned}$$

Concentrating on a given ${{\overline{\lambda }}}_I$, formal integration amounts to selecting, in the expansion of $\exp (\sum _{I' {\mathop {\subset }\limits ^{\circ }} J} \lambda _{I'} {{\overline{z}}}_{I'})$, precisely the terms involving the monomial ${{\overline{z}}}_I$, which by inspection come with an overall factor $\sum _{\pi \in {{\mathcal {P}}}(I)} \lambda _{\pi }$.

Though this is a formal integral, if the index set J is finite the result is a polynomial in $({{\overline{\lambda }}}_i)_{i\in J}$ and $(\lambda _I)_{I {\mathop {\subset }\limits ^{\circ }} J}$, and can be evaluated for “numerical” arguments. As established in the previous section, if $b_i$ are Bernoulli variables then

$$\begin{aligned} Z={{\mathbb {E}}}\left( \prod _{i\in J} (1+e_ib_i)\right) =1+\sum _{I {\mathop {\subset }\limits ^{\circ }} J} e_i \sum _{\pi \in {{\mathcal {P}}}(I)} K_{\pi }(b_I),\end{aligned}$$

with notations as above. Thus, we may write

$$\begin{aligned} Z= \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi } \exp (- z_i {{\overline{z}}}_i) (1+e_iz_i)\right) \exp \left( \sum _{I {\mathop {\subset }\limits ^{\circ }} J} K(b_I) {{\overline{z}}}_I\right) .\end{aligned}$$

In the sequel we do not distinguish between the formal variables $({{\overline{\lambda }}}_i)_{i\in J}$, $(\lambda _I)_{I {\mathop {\subset }\limits ^{\circ }} J}$ and their embodied counterparts $(e_i)_{i\in J}$, $(K(b_I))_{I {\mathop {\subset }\limits ^{\circ }} J}$, which we shall use in the formulæ.

We can turn the crank of Feynman graphs and rules as recalled in Appendix B and subsection B.1. Writing

$$\begin{aligned} \prod _{i\in J} (1+e_iz_i)=\exp \sum _{i\in J}\sum _{k\ge 1} \frac{(-1)^{k-1}(k-1)!e_i^kz_i^k}{k!} \end{aligned}$$

we infer that

$$\begin{aligned} \log Z = \sum _G w(G) \end{aligned}$$

where the sum is over connected bicolored graphs with white and black vertices whose edges carry a type $i\in J$, the type of the edges at a white vertex being all the same (constraint $\circ$) and at a black vertex all different (constraint $\bullet$). Each white vertex with k edges of type $i\in J$ contributes a factor $(-1)^{k-1}(k-1)!e_i^k$ to w(G). Each black vertex with edges whose types build a subset $I {\mathop {\subset }\limits ^{\circ }} J$ contributes a factor $K_I$ to w(G). There a an additional factor $1/ |\text {Aut}\, G|$ in w(G). The two constraints $(\circ )$ and $(\bullet )$ allow to “transfer” the types of edges to the white vertices and to consider connected bicolored graphs with white and black vertices, the white vertices carrying a tag $i\in J$ and the tags of white vertices connected to a black vertex being different.

We give some examples, assuming that J is a set of integers: – Examples: A white vertex of order 4 associated to site $i=3$, with weight $(-1)^{4-1}(4-1)!e_3^4=-6e_3^4$, a white vertex of order 3 associated to site 7, with weight $(-1)^{3-1}(3-1)!e_7^3=2e_7^3$ and a black vertex connected to sites 1, 3, 9, 5 with weight $K_4(b_1,b_3,b_9,b_5)$:

– Example: A diagram

with weight (reading more or less from left to right, in that simple case symmetries are “local” on the graph)

$$\begin{aligned}{} & {} \frac{1}{2!}K_{3}^2 (-1)^{2-1}(2-1)!e_6^2 \frac{1}{2!} K_{3,6}^2 (-1)^{6-1}(6-1)!e_3^6 K_{3,6}K_{3,6,7}\\{} & {} \quad (-1)^{3-1}(3-1)!e_6^3 (-1)^{2-1}(2-1)!e_7^2 K_{3,6,7}(-1)^{1-1}(1-1)!e_3, \end{aligned}$$

where $K_{3}$, $K_{3,6}$, $K_{3,6,7}$ are a shorthand notation for the cumulants $K_1(b_3)$, $K_2(b_3,b_6)$ and $K_3(b_3,b_6,b_7)$, respectively, a convention already used above.

The weight reduces to $-60 e_3^7e_6^5e_7^2K_3^2K_{3,6}^3K_{3,6,7}^2$.

From now on, we could reproduce with very little changes the discussion leading to the continuum limit, to which only tree diagrams contribute. – Here is an example of a tree:

the computation of whose weight is left to the reader. We pause for a moment to compare with the other (call it chromatic) graphical description of $\log Z$ given in Sect. 2.4.

For this, we need to rewrite some previous formulæ, in particular Eqs. (34) and (35). In the chromatic description, instead of summing over unlabeled graphs H and then over labelings by (distinct) elements of $J:=\{1,\cdots ,N\}$, we can as well sum over graphs G whose white vertices are labeled by $\{1,\cdots ,N\}$. If G is such a graph, define ${{\mathfrak {w}}}(G):=\prod _{\bullet }K(b_\bullet )\prod _{\hbox { edges of}\ G}e_i$, so that, if G is obtained from an unlabeled graph H via the labeling ${{\mathcal {L}}}$, ${{\mathfrak {w}}}(G)=w({{\mathcal {L}}})$. A moment thinking shows that

$$\begin{aligned} \frac{1}{|\text {Aut}\, H|} \sum _{{{\mathcal {L}}}\in Lab(H)}w({{\mathcal {L}}})= \sum _{G} \frac{1}{|\text {Aut}\, G|} {{\mathfrak {w}}}(G) \end{aligned}$$

where the sum on the right-hand side is over the distinct graphs G that can be obtained by labeling the white vertices of H, and $\text {Aut}\, G$ is the group of automorphisms of G respecting the labeling, which may be smaller than $\text {Aut}\, H$ because several labelings of H may induce the same G, as for instance in the example (38). As all white labels of G are distinct, the automorphism group of G is in fact very easy to describe: saying that two black vertices are equivalent if they are connected to the same set of white vertices, $\text {Aut}\, G$ is the group of permutations of equivalent black vertices. The definition of the operation $^\bullet$ for G can be copied on that of $H^\bullet$ and clearly $G^\bullet =H^\bullet$. Thus, the complete contribution of G to the free energy $\log Z$ in the chromatic description is $\frac{\mu (G^\bullet )}{|\text {Aut}\, G|} {{\mathfrak {w}}}(G)$.

The graphs G we have used for the Feynman graph description have a tag, an element of $J=\{1,\cdots ,N\}$, assigned to each white vertex, but this is not a labeling of the white vertices in general because several white vertices with the same tag are allowed.² Consequently, their automorphism group is more complicated to describe in general because it may permute white vertices as well. But the definition of the monomial ${{\mathfrak {w}}}(G)$ carries over without changes for Feynman graphs. On top of that, a white vertex of order k contributes a multiplicative factor $(-1)^{k-1}(k-1)!$. Defining $G^\circ$ as the collection of white vertices of G with their pending edges (i.e., one just removes the black vertices from the picture) and $\eta (G^\circ ):= \prod _{\text {white vertices of } G}(-1)^{k-1}(k-1)!$, we can put things together and write the complete contribution of G to the free energy $\log Z$ in the Feynman graph description as $\frac{\eta (G^\circ )}{|\text {Aut}\, G|} {{\mathfrak {w}}}(G)$.

Working as above with (connected) graphs whose white vertices are tagged, let us denote by ${{\mathcal {C}}}_C$ (resp. ${{\mathcal {C}}}_F$) the class of graphs involved in the chromatic (resp. Feynman graph) expansion of $\log Z$. We have just seen that ${{\mathcal {C}}}_C\subset {{\mathcal {C}}}_F$: the chromatic graphical description which is tailored for the problem at hand and is more economical because there are less graphs to consider. We have

$$\begin{aligned} \log Z = \sum _{G \in {{\mathcal {C}}}_C} \frac{\mu (G^\bullet )}{|\text {Aut}\, G|} {{\mathfrak {w}}}(G)=\sum _{G \in {{\mathcal {C}}}_F} \frac{\eta (G^\circ )}{|\text {Aut}\, G|} {{\mathfrak {w}}}(G). \end{aligned}$$

We may refine this identity using the obvious observation that for any finite collection (possibly with repetition) of non-empty finite subsets of J, say $(I_a)_{a\in A}$ there is a single graph G in ${{\mathcal {C}}}_C$ with black vertices indexed by the $I_a$, i.e., “black” weight $\prod _{a\in A} K_{I_a}$. In particular, a graph $G\in {{\mathcal {C}}}_C$ can be reconstructed from the sole knowledge of ${{\mathfrak {w}}}(G)$, leading to the identity

$$\begin{aligned} \frac{\mu (G^\bullet )}{|\text {Aut}\, G|}=\sum _{G'\in {{\mathcal {C}}}_F,\, {{\mathfrak {w}}}(G')={{\mathfrak {w}}}(G)} \frac{\eta ((G')^\circ )}{|\text {Aut}\, G'|}\quad \text {for } G\in {{\mathcal {C}}}_C.\end{aligned}$$

This result is perhaps more suggestive if one introduces a partial ordering on ${{\mathcal {C}}}_F$: for $G,G' \in {{\mathcal {C}}}_F$ say that $G'\succcurlyeq G$, or that $G'$ covers G if G is obtained from $G'$ by identifying some white vertices carrying the same tag. This is clearly a partial ordering on ${{\mathcal {C}}}_F$. The maximal elements are the trees, and the minimal elements are the elements of ${{\mathcal {C}}}_C$. If $G'\succ G$, G has less white vertices than $G'$ and G has more loops than $G'$. The graphs G and $G'$ have the same number of black vertices and the same number of edges, and in fact the equality ${{\mathfrak {w}}}(G)={{\mathfrak {w}}}(G)$ holds. In particular, given G there are finitely many $G'\succcurlyeq G$ and the previous identity rewrites

$$\begin{aligned} \frac{\mu (G^\bullet )}{|\text {Aut}\, G|}=\sum _{G'\in {{\mathcal {C}}}_F, G' \succcurlyeq G} \frac{\eta ((G')^\circ )}{|\text {Aut}\, G'|}\quad \text {for } G\in {{\mathcal {C}}}_C.\end{aligned}$$

(45)

The above argument gives a rigorous but indirect proof of this identity. A direct proof for trees is easy: if G in ${{\mathcal {C}}}_C$ is a tree, there is only G itself in the sum on the right-hand side (G is minimal and maximal for $\succcurlyeq$),³ and the $(-1)^{k-1}(k-1)!$s coming from complete graphs chromatic factors in $\mu (G^\bullet )$ match precisely with Feynman graph contributions for white vertices in $\eta ((G)^\circ )$. Note, however, that the “Feynman trees” allow for several white vertices with the same tag. They cover graphs with loops in the chromatic expansion. In both the chromatic and the Feynman graph expansion, loops are suppressed with respect to trees when the number of sites, |J|, grows without bounds. This is not in contradiction with the semiclassical expansion: for a fixed number of white vertices, the factor suppressing trees with multiple vertices carrying the same tag is just due to their rarity compared to trees with all white vertices carrying a different tag, and this matches precisely with the scaling in the covering formula. The partial order $\succcurlyeq$ on ${{\mathcal {C}}}_F$ suggests that a recursive approach might help to give a direct proof of (45) in general, but we have not tried to follow this path.

A final remark: the computation of $|\text {Aut}\, G|$ in the class ${{\mathcal {C}}}_F$ is NP-hard, just as is the computation of $\mu (G^\bullet )$ in the class ${{\mathcal {C}}}_C$, whereas the computation of $\eta (G^\circ )$ in ${{\mathcal {C}}}_F$ or $|\text {Aut}\, G|$ in ${{\mathcal {C}}}_C$ is trivial.

3.2 Thermodynamic Limits

One small thing that speaks in favor for the redundant description of $\log Z$ by Feynman graphs is that it generalizes plainly to related counting problems.

Suppose for instance that J is finite, that $e_i$ does not depend on i and $K_I$ depends only on the size |I| of $I\subset J$.⁴ We fix a family $(t_k)_{k\ge 1}$ of formal variables. We set $e_i:=e$ and $K_I=:t_k$ if $|I| =k$. For $I {\mathop {\subset }\limits ^{\circ }} J$ the number of partitions of I made of $m_1$ parts of size 1, $m_2$ parts of size 2, and so on with $\sum _{k\ge 1} km_k =|I|$ is $\frac{(\sum _{k\ge 1} km_k)!}{\prod _{k \ge 1} m_k! (k!)^{m_k}}$, leading to

$$\begin{aligned} Z=\sum _{{\underline{m}}} \frac{|J| !}{(|J|-\sum _{k\ge 1} km_k)!} \prod _{k\ge 1} \frac{1}{m_k!}\left( \frac{e^kt_k}{k!}\right) ^{m_k} \end{aligned}$$

where the sum is over sequences of integers ${\underline{m}}:=(m_1,m_2,\cdots )$. The point is that the combinatorics is precisely recovered in a formal Gaussian integral as

$$\begin{aligned} Z = \int \frac{d{{\overline{z}}}\wedge \text {d}z}{2i\pi \hbar } \exp \left( - \frac{z{{\overline{z}}}}{\hbar }\right) \, (1+ez)^{|J|} \exp \sum _{1 \le k \le |J|} \frac{t_k}{k!} \left( \frac{{\overline{z}}}{\hbar }\right) ^k. \end{aligned}$$

Notice that $\hbar$, whether a formal variable or a numerical value, plays a purely spectator role in this formula. The expansion of Z in terms of Feynman graphs would follow straightforwardly.

We use rescaled variables $\Phi _k=:(|J|)^{k-1} t_k$ for $k \ge 1$. Choosing $\hbar :=(|J|)^{-1}$ and precising the variables involved in Z we are led to

$$\begin{aligned} Z_{|J|}(e,\Phi _\bullet ) = \int \frac{d{{\overline{z}}}\wedge \text {d}z}{2i\pi (|J|)^{-1}} \exp \frac{1}{(|J|)^{-1}}\left( - z{{\overline{z}}}+\log (1+ez) + \sum _{k \ge 1 } \frac{\Phi _k}{k!} {{\overline{z}}}^k\right) .\end{aligned}$$

Thus, $(|J|)^{-1}$ plays the role that $\hbar$ plays in the general discussion of A.3 and B and letting |J| grow without bounds with $\Phi _\bullet :=(\Phi _k)_{k\ge 1}$ fixed we obtain that in the thermodynamic limit

$$\begin{aligned} \lim _{|J| \rightarrow \infty } \frac{\log Z_{|J|}(e,\Phi _\bullet )}{|J|} =F^*(e,\Phi _\bullet ) \end{aligned}$$

where $F^*(e,\Phi _\bullet )=-g^*h^*+\log (1+eg^*) +\sum _{k\ge 1} \Phi _k \frac{{h^*}^k}{k!}$ with $(g^*,h^*)$ solving the equations $h=\frac{e}{1+eg}$ and $g=\sum _{k \ge 1 } \Phi _k \frac{h^{k-1}}{(k-1)!}$. The formal power series g (resp. h) is what we denoted by U (resp. ${{\overline{V}}}$) in the general discussion of subsection A.3 and subsection B.1.

As a slight generalization of this extreme case, suppose that J is finite and $J=\cup _{a\in A} J_a$ is a partition of J indexed by some set A. Impose that $e_i$ is the same for all is in a given $J_a$, and write it as $e_a$. Fix a family $(t_k)_{k\ge 1}$ where each $t_k$ is a symmetric function on $A^k$ and for $I {\mathop {\subset }\limits ^{\circ }} J$ set $K_I=t_k(a_1,\cdots ,a_k)$ if $I=\{i_1,\cdots ,i_k\}$ has size k and $i_l\in J_{a_l}$ for $l=1,\cdots ,k$. The extreme case is recovered when A is a singleton. The counting of partitions in the extreme case generalizes straightforwardly and leads to the integral representation

$$\begin{aligned} Z = \int \left( \prod _{a\in A} \frac{d{{\overline{z}}}_a\wedge \text {d}z_a}{2i\pi \hbar } \exp (- z_a {{\overline{z}}}_a/\hbar ) (1+e_az_a)^{|J_a|} \right) \exp {\overline{{\mathcal {L}}}}_{|J|}({{\overline{z}}}_\bullet )\end{aligned}$$

where

$$\begin{aligned} {\overline{{\mathcal {L}}}}_N({{\overline{z}}}_\bullet ):= \sum _{1 \le k \le N} \frac{1}{\hbar ^k k!}\sum _{a_1,\cdots ,a_k\in A} t_k(a_1,\cdots ,a_k){{\overline{z}}}_{a_1}\cdots {{\overline{z}}}_{a_k}.\end{aligned}$$

Again $\hbar$ is a spectator role in this representation. We use rescaled variables $\Phi _k=:(|J|)^{k-1} t_k$ for $k \ge 1$. Choosing $\hbar :=(|J|)^{-1}$ and precising the variables involved in Z we are led to

$$\begin{aligned}{} & {} Z_{|J|}(e_\bullet ,\Phi _\bullet )=\int \prod _{a\in A} \frac{d{{\overline{z}}}_a\wedge \text {d}z_a}{2i\pi (|J|)^{-1}} \exp \frac{1}{(|J|)^{-1}}\\{} & {} \quad \times \left( \sum _{a\in A} (- z_a {{\overline{z}}}_a + p_a \log (1+e_az_a)) +{{\overline{L}}}_{|J|} ({{\overline{z}}}_\bullet )\right) ,\end{aligned}$$

where $p_a:=\frac{|J_a|}{|J|}$ for $a\in A$ and

$$\begin{aligned} {{\overline{L}}}_N({{\overline{z}}}_\bullet ):= \sum _{1 \le k \le N} \frac{1}{k!}\sum _{a_1,\cdots ,a_k\in A} \Phi _k(a_1,\cdots ,a_k){{\overline{z}}}_{a_1}\cdots {{\overline{z}}}_{a_k}.\end{aligned}$$

Thus, there is a semiclassical expansion in powers of $(|J|)^{-1}$, with A, $\Phi _\bullet$ and $(p_a)_{a\in A}$ fixed. The first contribution, proportional to |J| is given by the saddle point $F^*(e_\bullet ,\Phi _\bullet )$ where $F^*(e_\bullet ,\Phi _\bullet )=-\sum _{a\in A} g_a^*h_a^*+\sum _{a\in A} p_a\log (1+e_ag_a^*) +{{\overline{L}}}_\infty (h^*_\bullet )$ with $(g_a^*,h_a^*)_{a\in A}$ solving the equations $h_a=\frac{e_a}{1+e_ag_a}$ and $g_a=\frac{\partial {{\overline{L}}}}{\partial {{\overline{z}}}_a}(h_\bullet )$, $a\in A$. Thus, letting |J| grow without bounds with A, $\Phi _\bullet$ and $(p_a)_{a\in A}$ fixed (this might require taking |J| along some subsequence) we obtain

$$\begin{aligned} \lim _{|J| \rightarrow \infty } \frac{\log Z_{|J|}(e_\bullet ,\Phi _\bullet )}{|J|} =F^*(e_\bullet ,\Phi _\bullet )\end{aligned}$$

The full semiclassical expansion is valid only for $(p_a)_{a\in A}$ fixed, but this equation for the dominant contribution holds if the $(p_a)_{a\in A}$s are allowed to depend on |J| with corrections $o((|J|)^{-1})$ and $|J| \rightarrow \infty$ without restrictions.

This formula is a special case of Proposition 5 as we explain now. Take $(f_k)_{k\ge 1}$ a sequence of symmetric integrable functions, $f_k:[0,1]^k\rightarrow {{\mathbb {R}}}$ and consider the functional

$$\begin{aligned} F(e,f_\bullet ,g,q):= & {} -\int _{[0,1]} g(x)\, q(x)\text {d}x+\int _{[0,1]} \log (1+e(x)g(x))\, \text {d}x \\ {}{} & {} +\sum _{k\ge 1}\int _{0<x_1<\cdots <x_k} f_k(x_1,\cdots ,x_k) \, q(x_1)\text {d}x_1\cdots \, q(x_k)\text {d}x_k,\end{aligned}$$

where e, g, q are plain functions. Note that the multiple integral could also be written as $\frac{1}{k!} \int _{[0,1]^k}f_k(x_1,\cdots ,x_k) \, q(x_1)\text {d}x_1\cdots \, q(x_k)\text {d}x_k$, because the diagonals (where several x’s coincide) do not contribute. The general formulæ come via the extremization of F with respect to the functions g and q. One way to discretize this problem is to partition [0, 1] with measurable subsets $(M_a)_{a\in A}$ with size $\int _{M_a}\, \text {d}x =p_a>0$ for $a\in A$ and take e, g, q as simple functions, explicitly

$$\begin{aligned}e(x)=\sum _a e_a \textbf{1}_{x\in M_a} \quad g(x)=\sum _a g_a \textbf{1}_{x\in M_a} \quad q(x)=\sum _a q_ap^{-1}_a \textbf{1}_{x\in M_a}.\end{aligned}$$

Taking $\Phi _\bullet$ as the average of $f_\bullet$ over rectangles, explicitly

$$\begin{aligned} \Phi _k(a_1,\cdots ,a_k):=\frac{1}{p_{a_1}\cdots p_{a_k}} \int _{M_{a_1}\times \cdots \times M_{a_k}} f_k(x_1,\cdots ,x_k) \,\text {d}x_1\cdots \, \text {d}x_k,\end{aligned}$$

(the diagonals where several $a_l$s may coincide do count in the discrete object $\Phi _\bullet$) we find that the discretized version of $F(e,f_\bullet ,g,q)$ is

$$\begin{aligned}-\sum _{a\in A} g_aq_a+\sum _{a\in A} p_a\log (1+e_ag_a)+\sum _{k \ge 1} \frac{1}{k!}\sum _{a_1,\cdots ,a_k\in A} \Phi _k(a_1,\cdots ,a_k)q_{a_1}\cdots q_{a_k},\end{aligned}$$

which coincides as announced with the thermodynamic limit functional above, to be extremized with respect to $(g_a,q_a)_{a\in A}$. If one feels uneasy with formal integrals in infinite dimensions and a semiclassical expansion with respect to the formal parameter $\hbar$, one might feel more comfortable with—still formal but—finite-dimensional integrals and a semiclassical expansion with respect to the actual number J of Bernoulli random variables. The general formulæ would then follow by a density argument via the approximation of the $K(b_I)$s, $I {\mathop {\subset }\limits ^{\circ }} J$ by step functions.

4 Classical SSEP and Free Probability

4.1 The Classical SSEP

The aim of this section is to recall the definition of the classical SSEP and its large deviation function. Results are taken from the SSEP literature [9, 10, 20] where further details may be found.

The classical SSEP is a time continuous Markov chain describing particles moving along a finite 1D lattice, with sites indexed by $i=1,\cdots , N$ (the sites i and $i+1$ are adjacent). Let $\tau _i$ be the occupation number of the site i : $\tau _i=0$ (resp. $\tau _i=1)$ if the site i is unoccupied (resp. occupied). Each configuration is specified by the data of these occupancies $\{\tau _i;\ i=1,\cdots ,N\}$. The particles are allowed to jump on their nearby positions, to their left or right with equal probability rate, if the target position is unoccupied. The allowed local moves are therefore $[01]\rightarrow [10]$ or $[10]\rightarrow [01]$, while the local configurations [00] and [11] are frozen. Particles are injected and extracted at the two ends of the interval to drive the system out of equilibrium. The SSEP Markov matrix is defined accordingly to take these moves into account in a natural way. We denote by ${\mathbb {E}}_\textrm{ssep}$ the SSEP invariant measure (which is known to be unique).

One is interested in the continuum scaling limit $N\rightarrow \infty$, $x=i/N$ fixed, $0<x<1$. The occupation configurations $\{\tau _i\}$ then become continuous density profiles ${\mathfrak {n}}(x)$ on the interval [0, 1]. In this scaling limit, the densities at the two ends of the interval are fixed by the injection–extraction processes : ${\mathfrak {n}}(0)={n}_a$ and ${\mathfrak {n}}(1)={n}_b$ with $n_a$ and $n_b$ specified by the injection–extraction rates at the corresponding boundary. We shall use the convention $n_a=0$, $n_b=1$ (without loss of generality).

In the scaling limit, the mean density profile interpolates linearly between the two boundary densities : $\lim _{N\rightarrow \infty }{\mathbb {E}}_\textrm{ssep}[\tau _{i=[xN]}]=x$ (with the convention $n_a=0$, $n_b=1$). Fluctuations of the density profiles satisfy a large deviation principle [11, 12]. Namely,

$$\begin{aligned} {\mathbb {P}}\left[ \tau _{i=[xN]}\approx n(x)\right] \asymp _{N\rightarrow \infty } e^{-N \,I_\textrm{ssep}[n] } \end{aligned}$$

(46)

with $I_\textrm{ssep}$ the so-called large deviation rate function.

Let $F_\textrm{ssep}$ be the generating function of the density cumulants in the scaling limit. It is such that

$$\begin{aligned} {\mathbb {E}}_{\textrm{ssep}}\big [ e^{ \sum _i \tau _i h_i } \big ] \asymp _{N\rightarrow \infty } e^{N\,F_\textrm{ssep}[h] } ~, \end{aligned}$$

(47)

for $h_i=h(x=i/N)$ with h(x) a smooth function over the interval [0, 1]. If the occupation configuration $\{\tau _i\}$ approaches the density profiles ${\mathfrak {n}}(x)$, then $\sum _i \tau _i h_i$ approaches $N \int \! \text {d}x\, h(x){\mathfrak {n}}(x)$, so that $F_\textrm{ssep}$ can alternatively be defined by ${\mathbb {E}}_{\textrm{ssep}}\big [ e^{ N \int \! \text {d}x\, h(x){\mathfrak {n}}(x) } \big ] \asymp _{\varepsilon \rightarrow 0} e^{N F_\textrm{ssep}[h] }$. Assuming Eq. (47) to be true implies that the pth-order density cumulants scale like $N^{1-p}$ in the scaling limit.

As is well known, the two functions $I_\textrm{ssep}[n]$ and $F_\textrm{ssep}[h]$ are related by Legendre transform :

$$\begin{aligned} F_\textrm{ssep}[h] = \max _{n(\cdot )} \left[ \int _0^1\!\! \text {d}x\, h(x)n(x) - I_\textrm{ssep}[n]\right] . \end{aligned}$$

(48)

The large deviation function $F_\textrm{ssep}[h]$ has been given in the SSEP literature [9,10,11,12, 20] as the solution of an extremization problem (with the convention $n_a=0$, $n_b=1$) :

$$\begin{aligned}{} & {} F_\textrm{ssep}[h] = \max _{g(\cdot )}\, F[h;g], \quad F[h;g]:= \int _0^1 \!\!\! \text {d}x\left[ \log \big (1+g(x)e(x)\big ) - \log ({g'(x)})\right] , \nonumber \\ \end{aligned}$$

(49)

with $e(x)=e^{h(x)}-1$, as above, and g(x) solution of the nonlinear differential equation,

$$\begin{aligned} \big (1+g(x)\,e(x))\big )\, g''(x) = g'(x)^2\, e(x) ~, \end{aligned}$$

(50)

with boundary conditions $g(0)=0$ and $g(1)=1$ (for $n_a=0$, $n_b=1$). Equation(50) is the Euler–Lagrange equation for F[h; g] to be extremal with respect to variations of g.

Expanding $F_\textrm{ssep}[h]$ in power of h yields the first few density cumulants (up to sub-leading terms in 1/N) :

$$\begin{aligned}{} & {} {\mathbb {E}}_\textrm{ssep}[\tau _{i=[xN]}\tau _{i=[yN]}]^c= - N^{-1}\, x(1-y)~,\\{} & {} {\mathbb {E}}_\textrm{ssep}[\tau _{i=[xN]}\tau _{i=[yN]}\tau _{i=[zN]}]^c= N^{-2}\, x(1-2y)(1-z), \end{aligned}$$

for $0<x<y<z<1$. More generally, the nth-order SSEP cumulants scale as $N^{1-n}$ in the scaling limit, see Refs. [9, 10, 20]. We let $\psi ^\textrm{ssep}_n(x_1,\cdots ,x_n)$ be the scaled cumulants, at non-coincident points, defined as

$$\begin{aligned} \psi ^\textrm{ssep}_n(x_1,\cdots ,x_n):= \lim _{N\rightarrow \infty } N^{n-1}{\mathbb {E}}_\textrm{ssep}\left[ \tau _{[x_1N]}\cdots \tau _{[x_nN]}\right] ^c, \end{aligned}$$

(51)

for $x_k\in [0,1]$, all distincts. The limit is known to exist and to be smooth (actually piecewise polynomial) at non-coincident points.

4.2 The Quantum to Classical SSEP Correspondence

The aim of this section is to recall the definition of the quantum SSEP (Q-SSEP) as well as its relation with the classical SSEP. Results explained below are taken from Refs. [2, 3] where further details may be found.

The quantum SSEP is a model of stochastic quantum dynamics describing fermions hopping along a 1D chain, with sites indexed by $i=1,\cdots , N$ (the sites i and $i+1$ are adjacent). For an open chain in contact with external reservoirs at their boundaries, the quantum SSEP dynamics results from the interplay between unitary, but stochastic, bulk flows with dissipative, but deterministic, boundary couplings. The bulk flows induce unitary evolutions of the system density matrix $\rho _t$ onto $e^{-idH_t}\,\rho _t\,e^{idH_t}$ with Hamiltonian increments

$$\begin{aligned} dH_{t}=\sqrt{J}\,\sum _{j=1}^{N}\big (c_{j+1}^{\dagger }c_{j}\,dW_{t}^{j}+c_{j}^{\dagger }c_{j+1}\,d{\overline{W}}_{t}^{j}\big ), \end{aligned}$$

(52)

for a chain of length L, where $c_{j}$ and $c_{j}^{\dagger }$ are canonical fermionic operators, one pair for each site of the chain, with $\{c_{j},c_{k}^{\dagger }\}=\delta _{j;k}$, and $W_{t}^{j}$ and ${\overline{W}}_{t}^{j}$ are pairs of complex conjugated Brownian motions, one pair for each edge along the chain, with quadratic variations $dW_{t}^{j}d\overline{W}_{t}^{k}=\delta ^{j;k}\,dt$. The contacts with the external leads are modeled by Lindblad terms [18]. The resulting equation of motion for the system density matrix $\rho _t$ reads

$$\begin{aligned} d\rho _{t}=-i[dH_{t},\rho _{t}] -\frac{1}{2}[dH_{t},[dH_{t},\rho _{t}]]+{\mathcal {L}}_\textrm{bdry}(\rho _{t})dt, \end{aligned}$$

(53)

with $dH_t$ as above and ${\mathcal {L}}_\textrm{bdry}$ the boundary Lindbladian. The two first terms result from expanding the unitary increment $\rho _t \rightarrow e^{-idH_t}\,\rho _t\,e^{idH_t}$ to second order as indicated by Itô calculus. The third term codes for the dissipative boundary dynamics representing injection–extraction at the two the boundaries. We do not need here the precise expression for ${\mathcal {L}}_\textrm{bdry}$ but the latter can be found in the literature [2, 4].

The classical SSEP is embedded in Q-SSEP because the average Q-SSEP dynamics on density matrix diagonal in the particle number basis reduces to the classical SSEP. At each site along the chain, the full $|\bullet \rangle$ and empty $|\varnothing \rangle$ states, with, respectively, one and zero fermion, form a basis of states and diagonalize the particle number operators ${\hat{n}}_i:=c^\dag _ic_i$ (with eigenvalue 1 or 0). The states $|{\varvec{n}}\rangle$ diagonalizing all the particle numbers along the chain are thus indexed by the classical configurations ${\varvec{n}}=(\tau _1,\cdots ,\tau _N)$, with $\tau _i=0,1,$ the particle number at site i. A density matrix diagonal in this particle number basis specifies a probability measure on classical configurations since it can be written as $\rho _\textrm{diag} = \sum _{{\varvec{n}}} Q_{{\varvec{n}}}\, \Pi _{{\varvec{n}}}$, with $\Pi _{{\varvec{n}}}:=|{\varvec{n}}\rangle \langle {\varvec{n}}|$ the projector on the classical configuration ${\varvec{n}}$ and $Q_{{\varvec{n}}}$ a probability measure on ${\varvec{n}}$ : $\sum _{{\varvec{n}}} Q_{{\varvec{n}}}=1$, $Q_{{\varvec{n}}}\ge 0$.

By the Markov property of the Brownian motions, the average dynamics deduced from Eq. (53) defines a semi-group on the average density matrix ${\bar{\rho }}_t:={\mathbb {E}}[\rho _t]$, generated by a Lindbladian ${\mathcal {L}}_{\textrm{ssep}}$ :

$$\begin{aligned} \partial _t {\bar{\rho }}_t = {\mathcal {L}}_{\textrm{ssep}}({\bar{\rho }}_t). \end{aligned}$$

(54)

The latter is obtained by averaging the Q-SSEP stochastic equation of motion (53). It preserves diagonal density matrices and thus defines a flow—a Markov chain—on probability measures on classical configurations. Locally, ${\mathcal {L}}_{\textrm{ssep}}$ acts as follows :

$$\begin{aligned} {\mathcal {L}}_{\textrm{ssep}} (|\varnothing \varnothing \rangle \langle \varnothing \varnothing |)&=0 ~,\\ {\mathcal {L}}_{\textrm{ssep}}(|\varnothing \bullet \rangle \langle \varnothing \!\bullet \!|)&= J( -|\varnothing \bullet \rangle \langle \varnothing \!\bullet \!|+|\!\bullet \!\varnothing \rangle \langle \!\bullet \varnothing | )~,\\ {\mathcal {L}}_{\textrm{ssep}}(|\!\bullet \!\varnothing \rangle \langle \bullet \varnothing |)&= J( +|\varnothing \bullet \rangle \langle \varnothing \!\bullet \!|-|\!\bullet \!\varnothing \rangle \langle \!\bullet \varnothing | )~,\\ {\mathcal {L}}_{\textrm{ssep}}(|\!\bullet \!\bullet \rangle \langle \!\bullet \!\bullet \!|)&=0 ~. \end{aligned}$$

This coincides with the Markov matrix of the classical SSEP. This coincidence also holds for the boundary processes. Thus, the average Q-SSEP dynamics, when reduced to density matrices which are diagonal in the particle number basis, is that of the classical SSEP, as claimed.

As a consequence, the generating function of the steady fluctuations of the classical SSEP occupancies $\tau _j=0,1$ can be expressed as a quantum expectation value w.r.t. the steady averaged Q-SSEP density matrix :

$$\begin{aligned} {\mathbb {E}}_\textrm{ssep}\left[ e^{\sum _j h_j \tau _j} \right] = {\mathbb {E}}_\infty \Big [ \textrm{Tr}\big ( \rho \, e^{\sum _i h_i {\hat{n}}_i}\big )\Big ]= \textrm{Tr}\left( {\bar{\rho }}_\infty \, e^{\sum _i h_i {\hat{n}}_i}\right) ~, \end{aligned}$$

(55)

with ${\hat{n}}_i:=c^\dag _ic_i$ the quantum number operators, ${\bar{\rho }}_\infty :={\mathbb {E}}_\infty [\rho ]$ the mean Q-SSEP state, averaged w.r.t. the Q-SSEP steady measure denoted ${\mathbb {E}}_\infty$. In particular, the multi-point correlation functions of the occupation numbers in the classical SSEP coincide with the quantum expectation values of the number operators w.r.t. to steady averaged Q-SSEP density matrix.

To complete this correspondence we need to express the quantum expectation values of the number operators in terms of known data relative to the Q-SSEP steady measure. The latter is constructed by looking at the expectation values of the fermion two-point functions $G_{ij}:=\textrm{Tr}(\rho \, c^\dag _j c_i)$. It has been shown [2] that the leading cumulants of this matrix are those for which the matrix indices are organized along a loop, namely of the form ${\mathbb {E}}_\infty [ G_{i_1i_n}\cdots G_{i_3i_2}G_{i_2i_1}]^c$. These cumulants scale as $N^{1-n}$ in the scaling limit. We let $\psi ^{\#}_n(x_{1},\cdots ,x_{n})$ be the scaled cumulants at non-coincident points:

$$\begin{aligned} \psi ^{\#}_n(x_{1},\cdots ,x_{n}):=\lim _{N\rightarrow \infty } N^{n-1}\, {\mathbb {E}}_\infty \left[ G_{[x_1N][x_nN]}\cdots G_{[x_3N][x_2N]}G_{[x_2N][x_1N]} \right] ^c, \nonumber \\ \end{aligned}$$

(56)

for $x_k\in [0,1]$ all distincts. The limit is known to exist and to be smooth at non-coincident points. Equations characterizing the $\psi ^{\#}_n$’s have been written and analyzed in [3]. They were later shown [7] to be related to free cumulants, as we shall recall below.

Having introduced the main players, we can now state the relation between the non-coincident cumulants in the classical and quantum SSEP, see Ref. [2].

Proposition 6

For $0<x_k<1$ all different, we have

$$\begin{aligned} \psi _n^\textrm{ssep}(x_1,\cdots ,x_n) = (-1)^{n-1}\sum _{\sigma \in S_n/{\mathbb {Z}}_n} \psi ^{\#}_n(x_{\sigma _1},\cdots ,x_{\sigma _n}). \end{aligned}$$

(57)

The sum is over all permutations $\sigma$ modulo cyclic permutations. There are $(n-1)!$ terms in the sum.

The relation (57) essentially follows from Wick’s theorem, see Ref. [2]. Together with the link between the Q-SSEP cumulants and free probability that we shall describe below, Eq. (57) is the starting point of the following new construction of the classical SSEP large deviation function.

4.3 Non-coincident SSEP Cumulants from Free Cumulants

The aim of this section is, on the one hand, to relate the generating function of non-coincident SSEP cumulants to free probability and, on the other hand, to use this relation to derive a simple integral representation of this generating function.

In order to avoid confusion between the large deviation generating and rate functions as given in the previous SSEP literature [9,10,11,12, 20]—namely $F_\textrm{ssep}[h]$ and $I_\textrm{ssep}[n]$ defined above in Eqs. (47, 46)—and the ones that we shall determine using free cumulant techniques, we shall denote the latter with script letters—namely ${\mathfrak {F}}_\textrm{ssep}[h]$ and ${\mathfrak {I}}_\textrm{ssep}[n]$. We shall prove in Sect. 4.5 that they (of course) coincide.

The Q-SSEP steady measure, and hence the functions $\psi ^{\#}_n$, have been shown to be related to free cumulants [7].

Proposition 7

Let the interval [0, 1] equipped with the Lebesgue measure, denoted $\mu _L$, be viewed as a probability space. Let ${\mathbb {I}}_x=1_{[0,x]}$ be the indicator function of the interval [0, x] with $0<x<1$. We have $\mu _L({\mathbb {I}}_{x_1}\cdots {\mathbb {I}}_{x_n})=\min (x_1,\cdots ,x_n) =: x_1\wedge \cdots \wedge x_n.$

The loop-expectation values $\psi ^\#_n$ are identified as the free cumulants $R_n$ of those random variables with respect to the measure $\mu _L$. Namely ,

$$\begin{aligned} \psi ^\#_n(x_1,\cdots ,x_n) = R_n\big ({\mathbb {I}}_{x_1},\cdots ,{\mathbb {I}}_{x_n}\big ). \end{aligned}$$

(58)

The proof of this fact relied on a combinatorial analysis, see Ref. [7] for a set of characterizing equations for the steady cumulants $\psi ^\#_n$ [3]. See Ref. [14] for an alternative analytic proof based on analyzing the time evolution equations of the Q-SSEP cumulants.

As an illustration, we write the first few terms for low values of n, using the defining relation between moments and free cumulants [8, 21, 24, 27]. For $n=2,\, 3$, we have :

$$\begin{aligned} \psi ^\#_2(x,y)= & {} x\wedge y - xy ~,\\ \psi ^\#_3(x,y,z)= & {} (x\wedge y \wedge z) - (x\wedge y)z - (x\wedge z)y - (y\wedge z)x + 2 xyz ~, \end{aligned}$$

If $0<x<y<z<1$, we get

$$\begin{aligned} \psi ^\#_2(x,y)= & {} x(1-y) ~,\\ \psi ^\#_3(x,y,z)= & {} x(1-2y)(1-z) ~. \end{aligned}$$

For $n\le 3$ there is no difference between free and standard cumulants. The difference starts at $n\ge 4$. For $n=4$, we have, for any order between the points $x_j$:

$$\begin{aligned} \psi ^\#_4(x_1,x_2,x_3,x_4)\!=\! & {} (x_1\wedge x_2\wedge x_3\wedge x_4) \!-\! \big ((x_1\wedge x_2\wedge x_3)x_4 \!+\! \circlearrowleft _{[\mathrm {4\ terms\ in\ total}]}\big ) \\{} & {} +\big ((x_1\wedge x_2)x_3x_4 + \circlearrowleft _{[\mathrm {6\ terms\ in\ total}]}\big ) - 3x_1x_2x_3x_4\\{} & {} - \big ((x_1\wedge x_2)- x_1x_2)\big )\big ((x_3\wedge x_4)- x_3x_4)\big ) \\{} & {} - \big ((x_1\wedge x_4)- x_1x_4)\big )\big ((x_2\wedge x_3)- x_2x_3)\big ) ~. \end{aligned}$$

If we choose to order them on the segment [0, 1], i.e., $0<x_1<x_2<x_3<x_4<1$, we get:

$$\begin{aligned} \psi ^\#_4(x_1,x_2,x_3,x_4)= & {} x_1(1-3x_2-2x_3+5x_2x_3)(1-x_4) ~,\\ \psi ^\#_4(x_1,x_3,x_4,x_2)= & {} x_1(1-3x_2-2x_3+5x_2x_3)(1-x_4) ~,\\ \psi ^\#_4(x_1,x_3,x_2,x_4)= & {} x_1(1-4x_2-x_3+5x_2x_3)(1-x_4) ~. \end{aligned}$$

We observe that they depend on the ordering of the points on the line (for $n\ge 4$). Higher-order cumulants can similarly be computed recursively. Of course, the computation becomes more and more involved and we need to package it.

Let us introduce the generating function of the classical SSEP non-coincident cumulants—recall the latter are linked to the Q-SSEP expectation values via Eq. (57). It is defined by

$$\begin{aligned} {\mathfrak {F}}_0[a;v]:= \sum _{n\ge 1} \frac{v^n}{n!}\,\psi ^\textrm{ssep}_n[a],\quad \psi ^\textrm{ssep}_n[a]:= \int _0^1{\psi }^\textrm{ssep}_n(x_1,\cdots ,x_n) \prod _{k=1}^n a(x_k)\text {d}x_k, \nonumber \\ \end{aligned}$$

(59)

with ${\psi }^\textrm{ssep}_n$ the scaled non-coincident cumulants. Here, v is simply a counting parameter that we introduced for later convenience. To avoid confusion and to allow for future comparison with the formula obtained in the SSEP literature, we have used a specific notation for the SSEP large deviation function computed using free cumulant technique. We set ${\mathfrak {F}}_0[a]:={\mathfrak {F}}_0[a;v=1]$. Of course we have ${\mathfrak {F}}_0[a;v]={\mathfrak {F}}_0[av]$.

Lemma 8

The generating function ${\mathfrak {F}}_0$ of the classical SSEP non-coincident cumulants can be expressed in terms of free cumulants $R_n$ w.r.t. the Lebesgue measure as

$$\begin{aligned} {\mathfrak {F}}_0[a]= - \sum _{k\ge 1} \frac{(-1)^{n}}{n} R_n({\mathbb {I}}_{[a]}), \end{aligned}$$

(60)

with ${\mathbb {I}}_{[a]}:=\int \!\text {d}x\, a(x){\mathbb {I}}_x$. Equivalently, $\psi ^\textrm{ssep}_n[a] = (-1)^{n-1}(n-1)! \, R_n({\mathbb {I}}_{[a]})$.

Proof

This is a direct consequence of the classical-to-quantum SSEP correspondence (57) and the multi-linearity of the free cumulants which imply

$$\begin{aligned} \psi _n^\textrm{ssep}[a] = (-1)^{n-1}(n-1)! \, R_n({\mathbb {I}}_{[a]}). \end{aligned}$$

(61)

$\square$

The function ${\mathbb {I}}_{[a]}$ is a classical variable on [0, 1], equipped with the Lebesgue measure, and $R_n({\mathbb {I}}_{[a]})$ its nth free cumulants. For a single, hence commuting, variable, there is a simple relation between the generating function of the free cumulants and that of the moments [8, 21, 24, 27]. This relation goes through the resolvent. We shall now explain how this yields to an efficient way to compute the classical SSEP non-coincident cumulants and how it can be used it to derive a simple integral representation of ${\mathfrak {F}}_0[a]$.

Let $R_{[a]}(z)$ be the generating function of the free cumulants $R_n(-{\mathbb {I}}_{[a]})$ :

$$\begin{aligned} R_{[a]}(z):=\frac{1}{z}+\sum _{p\ge 1} R_p(-{\mathbb {I}}_{[a]})z^{p-1}. \end{aligned}$$

Let $G_{[a]}(z):=\mu _L( \frac{1}{ z+{\mathbb {I}}_{[a]} })$ be the generating function of the moments $\mu _L( -{\mathbb {I}}_{[a]}^p )$ :

$$\begin{aligned} G_{[a]}(z)=\sum _{p\ge 0} z^{-p-1}\,\mu _L( -{\mathbb {I}}_{[a]}^p ). \end{aligned}$$

From Refs. [8, 21, 24, 27], these two generating functions are inverse functions, i.e., $R_{[a]}(G_{[a]}(z))=z$, which reads

$$\begin{aligned} 1 + \sum _{p\ge 1} R_p(-{\mathbb {I}}_{[a]})G_{[a]}(z)^{p} = zG_{[a]}(z). \end{aligned}$$

(62)

Let $b(x):=-\int _x^1\!\text {d}y\, a(y)$, so that $b'(x)=a(x)$ with $b(1)=0$. As a function on [0, 1], we have ${\mathbb {I}}_{[a]}=-b$, since ${\mathbb {I}}_{[a]}(x)=\int _0^1\!\text {d}y\, a(y){\mathbb {I}}_{x<y}=\int _x^1\!\text {d}y\, a(y)$. Thus, $\varphi ( -{\mathbb {I}}_{[a]}^p) = \int _0^1\!\text {d}x\, b(x)^p=:\overline{b^p}$, and

$$\begin{aligned} G_{[a]}(z) = \int _0^1 \frac{\text {d}x}{z+{\mathbb {I}}_{[a]}(x)}= \int _0^1 \frac{\text {d}x}{z-b(x)}. \end{aligned}$$

(63)

We can turn the logic around and view x as a function of b: x(b) is then interpreted as the cumulative probability for b, i.e., $\text {d}x(b)$ is the probability density for the variable b. The fact that $x\in [0,1]$ is then natural.

Formulas (62,63) yield a simple recursive way to compute the free cumulants $R_p$ in terms of the moments $\overline{b^p}$ and hence the generating function of the classical SSEP cumulants. We have :

$$\begin{aligned} R_1\big (-{\mathbb {I}}_{[a]})= & {} \overline{b} ~,\nonumber \\ R_2\big (-{\mathbb {I}}_{[a]})= & {} \overline{b^2} -\overline{b}^2 ~,\nonumber \\ R_3\big (-{\mathbb {I}}_{[a]})= & {} \overline{b^3} - 3\,\overline{b^2}\,\overline{b} + 2\,\overline{b}^3 ~,\nonumber \\ R_4\big (-{\mathbb {I}}_{[a]})= & {} \overline{b^4} - 4\,\overline{b^3}\, \overline{b} + 10\, \overline{b^2}\, \overline{b}^2 - 2\, \overline{b^2}^2 + 5\,\overline{b}^4 ~,\ \textrm{etc}. \end{aligned}$$

(64)

Higher free cumulants can be computed recursively. Using Eq. (61) this yields the classical SSEP non-coincident cumulants. As a consequence :

Lemma 9

Let $b(x):=-\int _x^1\!\text {d}y\, a(y)$. We have the following integral representation of the generating function of non-coincident SSEP cumulants :

$$\begin{aligned} {\mathfrak {F}}_0[a] = \int _0^1 \!\text {d}x\, \log (z-b(x)) - z + 1,\quad \textrm{with}\ \int _0^1 \frac{\text {d}y}{z-b(y)} =1. \end{aligned}$$

(65)

Proof

Recall that ${\mathfrak {F}}_0[a;v]={\mathfrak {F}}_0[av]$. By definition, Eq. (59), we have

$$\begin{aligned} v\partial _v {\mathfrak {F}}_0[a;v] = - \sum _{k\ge 1} v^{n} R_n(-{\mathbb {I}}_{[a]}) = 1- v R_{[a]}(v), \end{aligned}$$

with $R_{[a]}(z){:=}\frac{1}{z}+\sum _{p\ge 1} \kappa _p(-{\mathbb {I}}_{[a]})z^{p-1}$ as above. Thus, the relation $R_{[a]}(G_{[a]}(z)){=}z$ becomes

$$\begin{aligned} 1 - v\partial _v {\mathfrak {F}}_0[a;v] = vz, \quad \textrm{with}\ v=\int _0^1 \frac{\text {d}y}{z-b(y)}. \end{aligned}$$

(66)

We simply have to check that the function (65) is indeed solution of the above equation, with the appropriate boundary condition. Assuming ${\mathfrak {F}}_0[a]$ given by the r.h.s. of (65), we have

$$\begin{aligned} {\mathfrak {F}}_0[a;v] = \int _0^1 \!\text {d}x\, \log [v(z-b(x))] - vz + 1, \end{aligned}$$

with $\int _0^1 \frac{\text {d}y}{z-b(y)} =v$. Computing its derivative with respect to v, we get (using that $z=z[b,v]$ is actually a function of v and b)

$$\begin{aligned} v\partial _v {\mathfrak {F}}_0[a;v] = 1-vz + v\left( \frac{\partial z}{\partial v}\right) \left( \int _0^1\frac{\text {d}x}{z-b(x)} -v\right) . \end{aligned}$$

The last term vanish by definition of $z=z[b,v]$ and we get $v\partial _v {\mathfrak {F}}_0[a;v] = 1-vz$, as required. It is easy to check that the function (65) has the appropriate behavior at small v. $\square$

This representation can also be used to recursively compute ${\mathfrak {F}}_0[a;v]$. Equation (66) can alternatively be written as

$$\begin{aligned} v\partial _v{\mathfrak {F}}_0[a;v] = 1 - vz,\quad \textrm{with} \quad v= G_{[a]}(z) ~. \end{aligned}$$

(67)

The last relation, $v=G_{[a]}(z)$, determines z as a function of v, recursively :

$$\begin{aligned} vz(v)= & {} 1 + \overline{b}\,v +(\overline{b^2}-\overline{b}^2)\,v^2 + (\overline{b^3} - 3\, \overline{b^2}\, \overline{b} + 2\,\overline{b}^3)\,v^3\\{} & {} + (\overline{b^4} - 4\,\overline{b^3}\, \overline{b} + 10\, \overline{b^2}\, \overline{b}^2 - 2\, \overline{b^2}^2 + 5\,\overline{b}^4 )\,v^4 +\cdots ~, \end{aligned}$$

This is of course the generating function of the free cumulants of b, see Eq. (64). This is a very efficient way to compute the multi-point SSEP cumulants at non-coincident points.

4.4 The Classical SSEP Large Deviation Function from Free Probability

Once the generating function ${\mathfrak {F}}_0[a]$ of non-coincident SSEP cumulants has been identified, the SSEP large deviation generating function ${\mathfrak {F}}_\textrm{ssep}[h]$ can be computed using the variational principle established in the previous Sects. 2 & 3:

$$\begin{aligned} {\mathfrak {F}}_\textrm{ssep}[h]= \max _{g(\cdot );q(\cdot )} \left[ \int _0^1 \!\!\! \text {d}x\big [ \log \big (1+g(x)(e^{h(x)}-1)\big ) - q(x)g(x)\big ] + {\mathfrak {F}}_0[q]\right] ,\nonumber \\ \end{aligned}$$

(68)

with ${\mathfrak {F}}_0[q]$ defined in Eq. (65). Since the rate function ${\mathfrak {I}}_\textrm{ssep}$ is the Legendre transform of the large deviation generating function, we have

Corollary 10

$$\begin{aligned} {\mathfrak {I}}_\textrm{ssep}[n]{} & {} =\max _{g(\cdot ), q(\cdot )} \left( \int _0^1\!\! \text {d}x \, \left[ n(x)\log \left( \frac{n(x)}{g(x)}\right) +(1-n(x))\log \left( \frac{1-n(x)}{1-g(x)}\right) +q(x)g(x) \right] -{\mathfrak {F}}_0[q]\right) .\nonumber \\ \end{aligned}$$

(69)

Proof

Using Eq. (68) and ${\mathfrak {I}}_\textrm{ssep}[n]=\max _{h(\cdot )} \left( \int _0^1 \!\! \text {d}x\, h(x)n(x) - {\mathfrak {F}}_\textrm{ssep}[h]\right)$, we write

$$\begin{aligned} {\mathfrak {I}}_\textrm{ssep}[n]=\max _{h(\cdot )} \left( \int _0^1 \!\! \text {d}x\, \left[ h(x) n(x) - \log \big (1+g(x)e(x)\big ) + q(x)g(x)\right] - {\mathfrak {F}}_0[q] \right) , \nonumber \\ \end{aligned}$$

(70)

with $e(x)=e^{h(x)}-1$. Here g and q are determined as the functions for which ${\mathfrak {F}}_\textrm{ssep}[h]$ takes its maximal value,

$$\begin{aligned} q(x)=\frac{e(x)}{1+g(x)e(x)} \quad ,\quad g(x)=\frac{\delta {\mathfrak {F}}_0[q]}{\delta q(x)}. \end{aligned}$$

(71)

The maximum of (70) is attained for

$$\begin{aligned} h(x)=\log \left( \frac{n(x)(1-g(x))}{g(x)(1-n(x))}\right) , \end{aligned}$$

from which we get

$$\begin{aligned} 1+g(x)e(x)=\frac{1-g(x)}{1-n(x)},\quad q(x)=\frac{n(x)}{g(x)}-\frac{1-n(x)}{1-g(x)}. \end{aligned}$$

(72)

Inserting this back into the expression the rate function yields

$$\begin{aligned} {\mathfrak {I}}_\textrm{ssep}[n]=\int _0^1 \text {d}x \, \left[ n \log \left( \frac{n}{g}\right) +(n-1)\log \left( \frac{1-g}{1-n}\right) +qg \right] -{\mathfrak {F}}_0[q]. \end{aligned}$$

The two conditions (71) or (72) can be relaxed in writing this expression as a maximization problem (and simplifying the expression) as in Eq. (69). Indeed, one can check that the extremization condition (69) yields the same conditions for g and q as in (71) or (72). $\square$

4.5 Equivalence with the Previously Known Formulation

Finally, we present an explicit check that our new formula (68) for the SSEP large deviation function is identical to the previously known formula (49).

To prove this equivalence, we first formulate differently, but equivalently, the variational problem (49), as follows :

$$\begin{aligned} F_\textrm{ssep}[h]{} & {} = \max _{g(\cdot );f(\cdot )} {\widehat{F}}[h;f,g], {\widehat{F}}[h;f,g]:= \int _0^1 \!\!\! \text {d}x\big [ \log \big (1+g(x)\, e(x)\big ) - f(x)g(x)\big ] + V[f], \end{aligned}$$

where the functional V[f] is defined by

$$\begin{aligned} V[f]:= \int _0^1 \text {d}x\, \log \big (w-\ell (x)\big ) - w + 1,\quad \textrm{with}\ \int _0^1 \frac{\text {d}y}{w-\ell (y)} =1, \end{aligned}$$

(73)

with $\ell (x):=-\int _x^1 \text {d}y f(y)$, so that $\ell '(x)=f(x)$ with $\ell (1)=0$. We view w as a function of $\ell$ through the constraint $\int _0^1 \frac{\text {d}y}{w-\ell (y)} =1$.

To prove the equivalence between the two variational problems, we first have to compute the functional derivative of V[f]. Chain rule implies

$$\begin{aligned} \frac{\delta V[f]}{\delta f(x)} = -\int _0^1\left( \frac{\delta \ell (y)}{\delta f(x)}\right) \left( \frac{\text {d}y}{w-\ell (y)}\right) + \left( \frac{\delta w}{\delta f(x)}\right) \,\left( \int _0^1 \frac{\text {d}y}{w-\ell (y)} -1\right) . \end{aligned}$$

The second term vanish due to the relation $\int _0^1 \frac{\text {d}y}{w-\ell (y)} =1$. The definition of $\ell$ as $\ell (x):=-\int _x^1 \text {d}y f(y)$ implies $\frac{\delta \ell (y)}{\delta f(x)}=-{\mathbb {I}}_{\{x>y\}}$. Thus,

$$\begin{aligned} \frac{\delta V[f]}{\delta f(x)} = \int _0^x \frac{\text {d}y}{w-\ell (y)}. \end{aligned}$$

(74)

The extremization conditions (49) read

$$\begin{aligned} f(x) = \frac{e(x)}{1+e(x)g(x)},\quad g(x) = \frac{\delta V[f]}{\delta f(x)} = \int _0^x \frac{\text {d}y}{w-\ell (y)}. \end{aligned}$$

The relation $\int _0^1 \frac{\text {d}y}{w-\ell (y)} =1$ implies the boundary conditions $g(1)=1$. The last condition is equivalent to $1/g'(x)= w-\ell (x)$ with $g(0)=0$, and hence to $(1/g'(x))'= -\ell '(x)=-f(x)$, with $f(x)=\frac{e(x)}{1+e(x)g(x)}$, which is then equivalent to (50).

The last step consists in verifying that the extremum value coincide. Thanks to the extremum conditions, written as $g'(x)(w-\ell (x))=1$, and the boundary conditions on g and $\ell$, we have

$$\begin{aligned} \int _0^1 \!\text {d}x\, f(x)g(x)= \int _0^1 \!\text {d}x\, \ell '(x)g(x)= -\int _0^1 \!\text {d}x\, \ell (x)g'(x) = 1-w, \end{aligned}$$

so that

$$\begin{aligned} {\widehat{F}}[h;f,g]\vert _\textrm{ext} - F[h;g]\vert _\textrm{ext} = -\int _0^1 \!\text {d}x\, f(x)g(x) - w +1 =0 \end{aligned}$$

It is now clear that the two extremization problems (49) and (68) are equivalent, with the correspondence is $w\leadsto z$, $\ell \leadsto -{\mathbb {I}}_{[a]}$, so that ${\mathfrak {F}}_\textrm{ssep}[h] = F_\textrm{ssep}[h]$.

Appendices

Appendices

A Formalities

1.1 A.1 Formal Power Series

If R is any commutative ring with unit, S is an arbitrary index set and $\lambda _\bullet :=(\lambda _s)_{s\in S}$ a collection of variables, we denote by $R[\lambda _\bullet ]$, the ring of polynomials and by $R(\lambda _\bullet )$ the ring of formal power series in the variables $\lambda _\bullet$ with coefficients in R. We let $R(\lambda _\bullet )_{\ge 1}$ denote the ideal in $R(\lambda _\bullet )$ of formal power series with vanishing constant coefficient. The rings $R[\lambda _\bullet ]$ and $R(\lambda _\bullet )$ are again commutative rings with unit, so if J is a new arbitrary index set and $z_\bullet :=(z_i)_{i\in J}$, ${{\overline{z}}}_\bullet :=({{\overline{z}}}_i)_{i\in J}$ are new variables we may consider for instance $R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )$, the ring of formal power series in the variables $\lambda _\bullet$ with coefficients in the ring $R[z_\bullet ,{{\overline{z}}}_\bullet ]$ of polynomials in $z_\bullet ,{{\overline{z}}}_\bullet$. We note that $R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )\subset R(z_\bullet ,{{\overline{z}}}_\bullet ,\lambda _\bullet )$. Notice that in $R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )$ the “constant” coefficient is now a polynomial in $R[z_\bullet ,{{\overline{z}}}_\bullet ]$ and this polynomial vanishes for members of $R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )_{\ge 1}$. If F is any formal power series in one variable and $L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )\in R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )_{\ge 1}$ then the composition $F\circ L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )$ is well defined as an element of $R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )$. For instance, if $A_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )\in R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )$ then $A_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet ) e^{L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )} \in R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )$.

1.2 A.2 Formal Gaussian Integrals

From now on, the ground ring R is assumed to be a field of characteristic 0, say ${\mathbb {R}}$.

Our first aim is to make sense of

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) A_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet ) \end{aligned}$$

where $\hbar$ is yet another formal variable and $A_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )\in R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )$. The result of integration will be a element of $R(\hbar ,\hbar ^{-1},\lambda _\bullet )$ and in fact $A_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )$ may contain $\hbar$ explicitly (the notation is already heavy enough). It is useful to notice that contrary to the other formal variables involved, $\hbar$ can be specialized to a numerical value without impact on most of the discussions that follow (the semiclassical expansion below being an important exception) and then the result of integration is in $R(\lambda _\bullet )$.

We define the integral by term by term integration of the $\lambda _\bullet$-formal power series expansion of $A_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )$ so we are left with the task of defining

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) P(z_\bullet ,{{\overline{z}}}_\bullet ) \end{aligned}$$

for $P(z_\bullet ,{{\overline{z}}}_\bullet ) \in R[z_\bullet ,{{\overline{z}}}_\bullet ]$. Imposing linearity, it is enough to deal with monomials in $R[z_\bullet ,{{\overline{z}}}_\bullet ]$, i.e., expressions of the form $\prod _{i\in J} z_i^{m_i} {{\overline{z}}}_i^{{{\overline{m}}}_i}=:z_\bullet ^{m_\bullet } {{\overline{z}}}_\bullet ^{{{\overline{m}}}_\bullet }$ where $m_i,{{\overline{m}}}_i\in {\mathbb {N}}$ and $\sum _{i\in J} m_i+{{\overline{m}}}_i$, called the degree of the monomial, is finite. For such a monomial we set

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \prod _{i\in J} z_i^{m_i} {{\overline{z}}}_i^{{{\overline{m}}}_i}:= \prod _{i\in J} \delta _{m_i,{{\overline{m}}}_i}\hbar ^{m_i} m_i!,\end{aligned}$$

a formula copied from the honest integral over the complex plane

$$\begin{aligned} \int _{{\mathbb {C}}} \frac{d{{\overline{z}}}\wedge \text {d}z}{2i\pi \hbar } \exp (- z {{\overline{z}}}/\hbar ) z^m {{\overline{z}}}^{{\overline{m}}}=\delta _{m,{{\overline{m}}}}\hbar ^{m} m!,\end{aligned}$$

which holds for $m,{{\overline{m}}}\in {\mathbb {N}}$ and $\hbar$ a complex number with strictly positive real part.

Our main interest lies in the computation of

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar \end{aligned}$$

where $L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )\in R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )_{\ge 1}$. As $\exp L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar$ belongs to $R[z_\bullet ,{{\overline{z}}}_\bullet ]$ $(\lambda _\bullet )$ this is really a special case of the previous discussion.

We focus for a while on the case when the index set, J for the variables $(z_\bullet ,{{\overline{z}}}_\bullet )$ is fixed. We denote by S(J) the set indexing monomials in $(z_\bullet ,{{\overline{z}}}_\bullet )$, i.e.,

$$\begin{aligned} S(J):=\Bigg \{(m_i,{{\overline{m}}}_i)_{i\in J}, \, m_i,{{\overline{m}}}_i\in {\mathbb {N}} \text { for } i\in J,\, \sum _{i\in J} m_i+{{\overline{m}}}_i <+\infty \Bigg \}.\end{aligned}$$

The variables indexed by S(J) are denoted by $\Lambda _\bullet :=(\Lambda _{m_\bullet ,{{\overline{m}}}_\bullet })_{(m_\bullet ,{{\overline{m}}}_\bullet )\in S(J)}$ and our goal is the computation of

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar , \end{aligned}$$

where

$$\begin{aligned} L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet ):=\sum _{(m_\bullet ,{{\overline{m}}}_\bullet )\in S(J)} \Lambda _{m_\bullet ,{{\overline{m}}}_\bullet } \prod _{i\in J} \frac{z_i^{m_i}}{m_i!} \frac{{{\overline{z}}}_i^{{{\overline{m}}}_i}}{{{\overline{m}}}_i!}\in R[z_\bullet ,{{\overline{z}}}_\bullet ](\Lambda _\bullet )_{\ge 1}.\end{aligned}$$

The appearance of factorials in the denominator will simplify the forthcoming formulæ.

We call this special case “the” master integral. It contains all the information to deal with the general case. Indeed, if B is an arbitrary set indexing variables $\lambda _\bullet :=(\lambda _s)_{s\in B}$, any $L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )\in R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )_{\ge 1}$ can by expanded in monomials:

$$\begin{aligned} L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )=\sum _{(m_\bullet ,{{\overline{m}}}_\bullet )\in S(J)} \Lambda (B)_{m_\bullet ,{{\overline{m}}}_\bullet } \prod _{i\in J} \frac{z_i^{m_i}}{m_i!} \frac{{{\overline{z}}}_i^{{{\overline{m}}}_i}}{{{\overline{m}}}_i!}\end{aligned}$$

where each $\Lambda (B)_{m_\bullet ,{{\overline{m}}}_\bullet }$ belongs to $R(\lambda _\bullet )_{\ge 1}$. Notice, however, that the formal power series $\Lambda (B)_{m_\bullet ,{{\overline{m}}}_\bullet }$ are not arbitrary: a given $\lambda _\bullet$ appears in only finitely many $\Lambda (B)_{m_\bullet ,{{\overline{m}}}_\bullet }$ so that its coefficient in the $\lambda _\bullet$ expansion of $L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )$ is indeed a polynomial in $z_\bullet ,{{\overline{z}}}_\bullet$. Anyway,

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar \end{aligned}$$

is recovered from

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar \end{aligned}$$

by substitution, for each $(m_\bullet ,{{\overline{m}}}_\bullet )\in S(J)$ of the formal power series $\Lambda (B)_{m_\bullet ,{{\overline{m}}}_\bullet }$ for the formal variable $\Lambda _{m_\bullet ,{{\overline{m}}}_\bullet }$.

1.3 A.3 Semiclassical Expansion

As a warm-up exercise, the reader is invited to check that, as a formal integral where $z,{{\overline{z}}},u$ and ${{\overline{v}}}$ are formal variables,

$$\begin{aligned} \int \frac{d{{\overline{z}}}\wedge \text {d}z}{2i\pi \hbar } \exp (- (z+u)({{\overline{z}}}+{{\overline{v}}})/\hbar )=1.\end{aligned}$$

This simple identity, which can be read as translation invariance (independently over $z, {{\overline{z}}}$ so even if it is valid for its honest integral avatar in the complex plane, this does not reduce to translation invariance of the Lebesgue measure), plays an important role in the manipulation of (formal) Gaussian integrals. The reader should also check the corollaries

$$\begin{aligned}{} & {} \int \frac{d{{\overline{z}}}\wedge \text {d}z}{2i\pi \hbar } \exp (- z{{\overline{z}}}/\hbar )\exp (- ({{\overline{v}}}z/ +u{{\overline{z}}})/\hbar )=e^{u {{\overline{v}}}/\hbar },\\{} & {} \int \frac{d{{\overline{z}}}\wedge \text {d}z}{2i\pi \hbar } \exp (- z{{\overline{z}}}/\hbar )\exp (((\overline{\nu }-{{\overline{v}}})z +(\mu -u){{\overline{z}}}+{{\overline{\nu }}}u+\mu {{\overline{v}}}-u{{\overline{v}}})/\hbar )=e^{\mu {{\overline{\nu }}}/\hbar },\end{aligned}$$

whose expansion in the new formal variables $\mu ,{{\overline{\nu }}}$ yields a more general version of translation invariance

$$\begin{aligned} \int \frac{d{{\overline{z}}}\wedge \text {d}z}{2i\pi \hbar } \exp (- (z+u)({{\overline{z}}}+{{\overline{v}}})/\hbar ) (z+u)^m ({{\overline{z}}}+{{\overline{v}}})^{{{\overline{m}}}}= \delta _{m,{{\overline{m}}}}\hbar ^{m} m!,\end{aligned}$$

which also plays an important role in what follows.

We introduce collections $u_\bullet :=(u_i)_{i\in J}$, ${{\overline{v}}}_\bullet :=({{\overline{v}}}_i)_{i\in J}$ of formal variables and use translation invariance:

$$\begin{aligned}{} & {} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar \\{} & {} \quad = \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- (z_i+u_i)({{\overline{z}}}_i+{{\overline{v}}}_i)/\hbar ) \right) \\{} & {} \qquad \times \exp L_{\lambda _\bullet }(z_\bullet +u_\bullet ,{{\overline{z}}}_\bullet +{{\overline{v}}}_\bullet )/\hbar .\end{aligned}$$

We infer that if $U_\bullet :=(U_i)_{i\in J}$, ${{\overline{V}}}_\bullet :=(V_i)_{i\in J}$ are arbitrary elements of $R(\lambda _\bullet )_{\ge 1}$ then

$$\begin{aligned}{} & {} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar \\{} & {} \quad =\int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- (z_i+U_i)({{\overline{z}}}_i+{{\overline{V}}}_i)/\hbar ) \right) \\{} & {} \qquad \times \exp L_{\lambda _\bullet }(z_\bullet +U_\bullet ,{{\overline{z}}}_\bullet +{{\overline{V}}}_\bullet )/\hbar .\end{aligned}$$

We claim that for any given $L_{\lambda _\bullet }$ there are unique members $U_\bullet ^*$, ${{\overline{V}}}_\bullet ^*$ in $R(\lambda _\bullet )_{\ge 1}$ such that

$$\begin{aligned} \sum _{i\in J} (z_i+U_i^*) ({{\overline{z}}}_i+{{\overline{V}}}_i^*)-L_{\lambda _\bullet }(z_\bullet +U_\bullet ^*,{{\overline{z}}}_\bullet +{{\overline{V}}}_\bullet ^*)\end{aligned}$$

has an extremum in $z_\bullet$ and ${{\overline{z}}}_\bullet$ at 0. Indeed the extremum equations are

$$\begin{aligned} {{\overline{V}}}_i=\frac{\partial L_{\lambda _\bullet }}{\partial z_i} (U_\bullet ,{{\overline{V}}}_\bullet ) \qquad U_i=\frac{\partial L_{\lambda _\bullet }}{\partial {{\overline{z}}}_i} (U_\bullet ,{{\overline{V}}}_\bullet )\end{aligned}$$

for $i\in J$, and existence/uniqueness of $(U_\bullet ^*,{{\overline{V}}}_\bullet ^*)$ follow from a tedious but straightforward recursive argument on the degrees in the $(z_\bullet ^*,{{\overline{z}}}_\bullet ^*)$ of $L_{\lambda _\bullet }$.

Set $F^*:= -\sum _{i\in J} U_i^*{{\overline{V}}}_i^*+L_{\lambda _\bullet }(U_\bullet ^*,{{\overline{V}}}_\bullet ^*)$, a member of $R(\lambda _\bullet )_{\ge 1}$. Then,

$$\begin{aligned}{} & {} -\sum _{i\in J} (z_i+U_i^*) ({{\overline{z}}}_i+{{\overline{V}}}_i^*)+L_{\lambda _\bullet }(z_\bullet +U_\bullet ^*,{{\overline{z}}}_\bullet +{{\overline{V}}}_\bullet ^*)-F_{\lambda _\bullet }^*\\{} & {} \quad +\sum _{i\in J} z_i{{\overline{z}}}_i=: L_{\lambda _\bullet }^*(z_\bullet ,{{\overline{z}}}_\bullet )\end{aligned}$$

belongs to $R[z_\bullet ,{{\overline{z}}}_\bullet ](\lambda _\bullet )_{\ge 1}$ (this fact concerns the $\lambda _\bullet$-expansion) and involves only terms of degree $\ge 2$ in $(z_\bullet ,{{\overline{z}}}_\bullet )$. Consequently, a simple power counting argument shows that

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp L_{\lambda _\bullet }^*(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar \end{aligned}$$

belongs to $R(\hbar ,\lambda _\bullet )$ (no $\hbar ^{-1}$ involved) and that the constant term is 1. Putting everything together leads to

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp L_{\lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar =\exp F_{\lambda _\bullet }^*/\hbar +\sum _{n\in {\mathbb {N}}} \hbar ^n F_{\lambda _\bullet }^{(n)} \end{aligned}$$

where each $F_{\lambda _\bullet }^{(n)}$ belongs to $R(\lambda _\bullet )_{\ge 1}$. This $\hbar$-expansion is the formal version of the saddle point expansion. We shall soon see a diagrammatic interpretation of this result.

B Feynman Graphs and Rules

We turn to graphical rules allowing the computation (in principle) of the master integral.

Consider a $\Lambda _\bullet$-monomial in the expansion of $\exp L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar$. It comes with:

• A coefficient which is proportional (the coefficient is in R) to a $(z_\bullet ,{{\overline{z}}}_\bullet )$ monomial, and this monomial survives integration (gives a nonzero contribution) if and only if for each $i\in J$ the power of $z_i$ and ${{\overline{z}}}_i$ are equal, say $n_i$. In this case, we say that the given $\Lambda _\bullet$-monomial is balanced. Then, its integral is $\prod _{i\in J} n_i!$.

• A combinatorial denominator, a product of factorials. If $\Lambda _{m_\bullet ,{{\overline{m}}}_\bullet }$ appears to the power k in the monomial, it contributes a multiplicative factor $k!(\prod _{i\in J} m_i!{{\overline{m}}}_i!)^k$ to the combinatorial denominator. The k! comes from the expansion of the exponential function, and the other factor from the definition of $L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )$

We translate the first fact as follows. For $(m_\bullet ,{{\overline{m}}}_\bullet )\in S(J)$, we represent $\Lambda _{m_\bullet ,{{\overline{m}}}_\bullet } \prod _{i\in J} z_i^{m_i}{{\overline{z}}}_i^{{{\overline{m}}}_i}$ as a vertex with $m_i$ outgoing edges and ${{\overline{m}}}_i$ incoming edges carrying the tag i. Thus, a term in the expansion of $\exp L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar$ is represented as a collection of vertices with pending edges. We represent the integration as a black box in which all the edges meet:

• If the number of incoming edges and outgoing edges carrying the tag $i\in J$ are not equal, the integration yields 0.

• If both equal $n_i$, integration yields a factor $n_i!$. The obvious but crucial observation is: $n_i!$ is precisely the number of ways to pair the incoming edges of type i to the outgoing edges of type i, that is, the number of ways to organize the inside of the black box so that no pending edge remains. For each such choice of pairing, opening the black box reveals an oriented graph, in which each edges carries a type $i\in J$. Conversely, given such a graph, breaking each edge in two pending edges (keeping track of the type and the orientation) one reconstructs a term in the expansion of $\exp L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar$.

We stress that the graphs the appear in this graphical representation have their edges oriented and tagged but:

• They are not simple. For instance several edges with the same tag and orientation may join two vertices. Moreover they may have tadpoles.

• Neither the vertices nor the edges are labeled.

So we pause for a moment to establish the definitions. A (labeled) oriented graph with edges tagged by J is given by the following data: finite sets $V,E^\pm$, arbitrary maps $t^\pm :E^\pm \rightarrow J$, $a^\pm :E^\pm \rightarrow V$, and a bijection $b:E^+ \rightarrow E^-$ such that $t^- \circ t^-b=t^+$. Though there are slightly simpler (but equivalent) definitions, this one is well suited to make contact with the graphical representation of the master integral. If $v\in V$ and $e^\pm \in E^\pm$, we view v as a vertex, $e^+$ as an outgoing edge tagged with $t^+(e^+)$ pending at vertex $a^+(e^+)$ and $e^-$ as an incoming edge tagged with $t^-(e^+)$ pending at vertex $a^-(e^-)$ and the $(e^+,e^-)$ is an oriented edge (made by two pending edges) tagged with $i\in J$ if $e^-=b(e^+)$ and $t^+(e^+)=i$ (in which case $t^-(e^-)=i$ as well). The definitions ensure that, for each $i\in J$, $(t^+)^{-1}(i)$ and $(t^-)^{-1}(i)$ have the same size.

To $(V,E^\pm ,t^\pm ,a^\pm )$ (i.e., forgetting b for a moment), we may associate a balanced $\Lambda _\bullet$-monomial (one whose integral gives a nonzero result): if vertex v has $m_i$ outgoing pending edges and ${{\overline{m}}}_i$ incoming pending edges tagged with i, it contributes a factor $\Lambda _{m_\bullet ,{{\overline{m}}}_\bullet }$ to the $\Lambda _\bullet$-monomial. Then, b describes a wiring inside the integration black box.

Two graphs $(V_1,E^\pm _1,t^\pm _1,a^\pm _1,b_1)$ and $(V_2,E^\pm _2,t^\pm _2,a^\pm _2,b_2)$ are equivalent if there are bijections $\omega :V_1 \rightarrow V_2$ and $\phi ^\pm :E^\pm _1 \rightarrow E^\pm _2$ such that $t^\pm _2\circ \phi ^\pm = t^\pm _1$, $a^\pm _2 \circ \phi ^\pm =\omega \circ a^\pm _1$ and $b_2\circ \phi ^+=\phi ^- \circ b_1$. Clearly two equivalent graphs yield the same associated balanced $\Lambda _\bullet$-monomial. Notice that, tautologically, if $(V_1,E^\pm _1,t^\pm _1,a^\pm _1,b_1)$ and there are bijections $\omega :V_1 \rightarrow V_2$ and $\phi ^\pm :E^\pm _1 \rightarrow E^\pm _2$, then $(V_2,E^\pm _2,t^\pm _1\circ (\phi ^\pm )^{-1},\omega \circ a^\pm _1\circ (\phi ^\pm )^{-1},\phi ^- \circ b_1\circ (\phi ^+)^{-1})$ is an equivalent graph.

Take a graph $(V,E^\pm ,t^\pm ,a^\pm ,b)$. Consider triples $(\omega ,\phi ^\pm )$, where $\omega :V \rightarrow V$ and $\phi ^\pm :E^\pm \rightarrow E^\pm$ are bijections such that $t^\pm \circ \phi ^\pm =t^\pm$ and $a^\pm \circ \phi ^\pm =\omega \circ a^\pm$. Such triples form a group under composition. The conditions that $(\omega ,\phi ^\pm )$ must satisfy mean that one may

• Arbitrarily permute the pending edges at a given vertex if they have the same orientation and tag

• Arbitrarily permute vertices (with their pending edges) if they have the same environment, that is if for each orientation and tag they have the same number of pending edges of that orientation and tag.

Thus, the order of this group is seen to match exactly the combinatorial denominator that comes with the balanced $\Lambda _\bullet$-monomial associated to the graph.

Moreover by construction $(V,E^\pm ,t^\pm ,a^\pm ,b)$ and $(V,E^\pm ,t^\pm ,a^\pm ,\phi ^-\circ b \circ (\phi ^+)^{-1})$ are equivalent graphs. If $\phi ^-\circ b =b \circ \phi ^+$ the action of the triple $(\omega ,\phi ^\pm )$ fixes the graph. This condition defines a subgroup of the above group. This subgroup is (called) the automorphism group of the graph.

Forgetting about set theoretic issues, an unlabeled graph is informally an equivalence class of labeled graphs. More concretely, given the size m of V and n of $E^\pm$, we may take arbitrary bijections from V to $\llbracket 1,m \rrbracket$ (we define $\llbracket 1,0 \rrbracket :=\emptyset$) and from $E^\pm$ to $\llbracket 1,n \rrbracket$ to define an equivalent graph, so we may for instance define an unlabeled graph as a graph with vertex set $\llbracket 1,m \rrbracket$ and edge set $\llbracket 1,n \rrbracket$ for some (m, n), modulo equivalence.

Back to our graph counting problem. Given a balanced $\Lambda _\bullet$-monomial (one whose integral gives a nonzero result) of degree m in the expansion of $\exp L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar$, with degree 2n in $(z_\bullet ,{{\overline{z}}}_\bullet )$ we may take $V=\llbracket 1,m \rrbracket$ to label the m “$\Lambda _\bullet$” factors, and $E^\pm =\llbracket 1,n \rrbracket$ to label the n “$z_\bullet$” and “${{\overline{z}}}_\bullet$” factors. This defines the maps $t^\pm ,a^\pm$. The wiring inside the integration black box defines the bijection b and we end with a (labeled) oriented graph with edges tagged by J. In general, different wirings may yield equivalent graphs (i.e., the same unlabeled graph) and it is more economical to group systematically such equivalent wirings. By the above discussion, this replaces the combinatorial denominator by the size of the automorphism group. In what follows, and unless obvious from the context, the term “graph” means “unlabeled graph.” If G is such a graph, we denote by $\text {Aut}\, G$ its automorphims group, whose cardinal is $|\text {Aut}\, G|$.

To summarize, we have obtained the following result:

• The master integral has a graphical expansion in formal power series

  $$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar = \sum _G w(G) \end{aligned}$$

  where the sum is over all unlabeled oriented graphs whose edges carry a type $i\in J$.

• The weight w(G) of a graph is computed as follows:

  • The edges incident at a vertex define a vertex type $(m_\bullet ,{{\overline{m}}}_\bullet ) \in S(J)$ counting how many incoming and outgoing edges of each type are incident, leading to a factor $\Lambda _{m_\bullet ,{{\overline{m}}}_\bullet }\hbar ^{-1}$ in w(G).

  • Each edge contributes a factor $\hbar$ in w(G).

  • There is an overall factor $1/ |\text {Aut}\, G|$ in w(G).

We give a few examples. For concreteness we take J to be the standard alphabet $J:=\{a,b,\cdots ,y,z\}$ (of which we use only a subset in the examples!).

– Example: A vertex with 3 outgoing and 2 incoming edges of type f, and an incoming edge of type u. It is convenient to use a compact notation, additive or multiplicative for instance, and denote the corresponding coupling by $\Lambda _{3f,2{{\overline{f}}}+{{\overline{u}}}}$ (additive, chosen below) or $\Lambda _{f^3{{\overline{f}}}^2{{\overline{u}}}}$ (multiplicative):

– Example: A diagram

with weight

$$\begin{aligned} \frac{1}{2!}\Lambda _{2b,{{\overline{e}}}} \Lambda _{b+e,2{{\overline{b}}}+{{\overline{k}}}} \Lambda _{k,{{\overline{d}}}}\Lambda _{d+h,{{\overline{b}}}}\Lambda _{h,{{\overline{h}}}}^2.\end{aligned}$$

The symmetry number 1/2! comes from the two equivalent outgoing type b edges at the vertex on the left.

Letting $n_v(G)$, $n_e(G)$, $n_l(G)$ and $n_c(G)$ denote, respectively, the number of vertices, edges, loops and connected components⁵ of G, we observe that the overall power of $\hbar$ in w(G) is $n_e(G)-n_v(G)$, which is also $n_l(G)-n_c(G)$ by a standard topological relation. At this point, we may pause to ask why it is worth to bother going from the plain “black box expansion” version of the master integral to the “graphical” version: after all, one black box gives rise to a whole family of graphs. One of the advantages of the graphical representation is that it behaves very nicely from the combinatorial viewpoint: the weight w(G) factors nicely over connected components, so that

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )/\hbar = \exp \sum _G w(G) \end{aligned}$$

where now the sum is over all connected oriented graphs whose edges carry a type $i\in J$, the weight being computed as before. The power of $\hbar$ in w(G) is simply $n_l(G)-1$ if G is connected, and comparing with the semiclassical expansion in Sect. **A.3 we infer that for the master integral the $(z_\bullet ,{{\overline{z}}}_\bullet )$-extremum $F(J)_{\Lambda _\bullet }^*:=F(J)_{\Lambda _\bullet }^{(-1)}$ of $\sum _{i\in J} z_i{{\overline{z}}}_i-L(J)_{\Lambda _\bullet }(z_\bullet ,{{\overline{z}}}_\bullet )$, and the corrections $F(J)_{\Lambda _\bullet }^{(n)}$, $n=0,1,\cdots$ are given by

$$\begin{aligned} F(J)_{\Lambda _\bullet }^{(n)}=\hbar ^{-n}\sum _G w(G) \text { for } n \ge -1,\end{aligned}$$

where the sum is over connected oriented graphs with $n+1$ loops whose edges carry a type $i\in J$.

1.1 B.1 A Special Case

We turn to a special case—one that suffices for our main interest—and change notations accordingly. We denote by S and ${{\overline{S}}}$ arbitrary sets, and introduce formal variables ${{\overline{\lambda }}}_\bullet :=({{\overline{\lambda }}}_s)_{s\in S}$ and $\lambda _\bullet :=(\lambda _s)_{s\in {{\overline{S}}}}$. The formal Gaussian integral of interest is now

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp (L_{{{\overline{\lambda }}}_\bullet }(z_\bullet )+ {{\overline{L}}}_{{\lambda }_\bullet }({{\overline{z}}}_\bullet ))/\hbar ,\end{aligned}$$

where $L_{{{\overline{\lambda }}}_\bullet }(z_\bullet )$ and ${{\overline{L}}}_{\lambda _\bullet }({{\overline{z}}}_\bullet )$ belong to $R[z_\bullet ]({{\overline{\lambda }}}_\bullet )_{\ge 1}$ and $R[{{\overline{z}}}_\bullet ](\lambda _\bullet )_{\ge 1}$, respectively. Thus, the “function” that we integrate against the Gaussian measure splits as a product of a “holomorphic function” and an “anti-holomorphic function,” and we are really dealing with a special case of the general discussion.

The semiclassical expansion carries through. The splitting between $z_\bullet$ and ${{\overline{z}}}_\bullet$ leads to the extremum equations

$$\begin{aligned} {{\overline{V}}}_i=\frac{\partial L_{{{\overline{\lambda }}}_\bullet }}{\partial z_i} (U_\bullet ) \qquad U_i=\frac{\partial {{\overline{L}}}_{\lambda _\bullet }}{\partial {{\overline{z}}}_i} ({{\overline{V}}}_\bullet )\end{aligned}$$

with solution $(U_\bullet ^*,{{\overline{V}}}_\bullet ^*)$ and extremal value $F^*:= \sum _{i\in J} U_i^*{{\overline{V}}}_i^*-L_{{{\overline{\lambda }}}_\bullet }(U_\bullet ^*)-{{\overline{L}}}_{\lambda _\bullet }({{\overline{V}}}_\bullet ^*)$

There is also a (restricted) master integral version adapted to the splitting. The index set J for the variables $(z_\bullet ,{{\overline{z}}}_\bullet )$ is fixed but we consider only monomials purely in $z_\bullet$ or in ${{\overline{z}}}_\bullet$. We denote now by S(J) the set indexing monomials in $z_\bullet$ so

$$\begin{aligned} S(J):=\left\{ (m_i)_{i\in J}, \, m_i\in {\mathbb {N}} \text { for } i\in J,\, \sum _{i\in J} m_i<+\infty \right\} .\end{aligned}$$

The variables indexed by S(J) are denoted by ${{\overline{\Lambda }}}_\bullet :=({{\overline{\Lambda }}}_{m_\bullet })_{(m_\bullet )\in S(J)}$. The set indexing monomials in ${{\overline{z}}}_\bullet$ is denoted by ${{\overline{S}}}(J)$ and the variables it indexes by $\Lambda _\bullet$. We specialize the general graphical rules to compute the restricted master integral

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp (L(J)_{{{\overline{\Lambda }}}_\bullet }(z_\bullet )+ {{\overline{L}}}(J)_{{\Lambda }_\bullet }({{\overline{z}}}_\bullet ))/\hbar ,\end{aligned}$$

where

$$\begin{aligned} L(J)_{{{\overline{\Lambda }}}_\bullet }(z_\bullet ):=\sum _{m_\bullet \in S(J)} {{\overline{\Lambda }}}_{m_\bullet }\prod _{i\in J} \frac{z_i^{m_i}}{m_i!} \in R[z_\bullet ]({{\overline{\Lambda }}}_\bullet )_{\ge 1},\end{aligned}$$

and

$$\begin{aligned} {{\overline{L}}}(J)_{\Lambda _\bullet }({{\overline{z}}}_\bullet ):=\sum _{{{\overline{m}}}_\bullet \in {{\overline{S}}}(J)} \Lambda _{{{\overline{m}}}_\bullet } \prod _{i\in J} \frac{{{\overline{z}}}_i^{{{\overline{m}}}_i}}{{{\overline{m}}}_i!}\in R[{{\overline{z}}}_\bullet ](\Lambda _\bullet )_{\ge 1}.\end{aligned}$$

Instead of one general family of vertices, we are led to consider two special families of vertices. For $m_\bullet \in S(J)$, we represent ${{\overline{\Lambda }}}_{m_\bullet } \prod _{i\in J} z_i^{m_i}$ as a white vertex with $m_i$ outgoing edges carrying the tag i, while for ${{\overline{m}}}_\bullet \in {{\overline{S}}}(J)$ we represent $\Lambda _{{{\overline{m}}}_\bullet } \prod _{i\in J} {{\overline{z}}}_i^{{{\overline{m}}}_i}$ as a black vertex with ${{\overline{m}}}_i$ incoming edges carrying the tag i. Thus, there is an expansion of the restricted master integral as

$$\begin{aligned} \int \left( \prod _{i\in J} \frac{d{{\overline{z}}}_i\wedge \text {d}z_i}{2i\pi \hbar } \exp (- z_i {{\overline{z}}}_i/\hbar ) \right) \exp (L(J)_{{{\overline{\Lambda }}}_\bullet }(z_\bullet )+ {{\overline{L}}}(J)_{{\Lambda }_\bullet }({{\overline{z}}}_\bullet ))/\hbar = \sum _G w(G) \end{aligned}$$

where the sum is over all bicolored (white,black) graphs whose edges carry a tag $i\in J$. The weight w(G) of a graph is computed as before, though in this restricted case slightly simpler rules could be given to compute the automorphism group. Note that the colors of vertices allow to reconstruct the orientation, edges going from white to black vertices. The restricted master integral exponentiates as $\exp \sum _G w(G)$ where now the sum is over all connected bicolored graphs whose edges carry a tag $i\in J$.

We give a few examples. We take again J to be the standard alphabet $J:=\{a,b,\cdots ,y,z\}$.

– Example: A white vertex with 6 (outgoing) edges, 3 with tag f, 2 with tag y and 1 with tag a, with weight ${{\overline{\Lambda }}}_{a+3f+2y}$:

– Example: A black vertex with 5 (incoming) edges, 2 with tag r, 2 with tag l and 1 with tag u, with weight $\Lambda _{2{{\overline{l}}}+2{{\overline{r}}}+{{\overline{u}}}}$:

– Example: A diagram

with weight (reading more or less from left to right, in that simple case symmetries are “local” on the graph)

$$\begin{aligned} \frac{1}{2!}\Lambda _{{{\overline{f}}}}^2 \frac{1}{2!} {{\overline{\Lambda }}}_{2m} \Lambda _{{{\overline{m}}}+{{\overline{n}}}}^2 {{\overline{\Lambda }}}_{2a+2f+2n} \Lambda _{{{\overline{a}}}+{{\overline{k}}}} \Lambda _{2{{\overline{a}}}+{{\overline{k}}}} {{\overline{\Lambda }}}_{a+i+k} {{\overline{\Lambda }}}_{2k} \Lambda _{2{{\overline{i}}}+{{\overline{k}}}} {{\overline{\Lambda }}}_{i}.\end{aligned}$$