Abstract
We study the two-time distribution in directed last passage percolation with geometric weights in the first quadrant. We compute the scaling limit and show that it is given by a contour integral of a Fredholm determinant.
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1 Introduction
In this paper we will consider the so called two-time distribution in directed last-passage percolation with geometric weights. This last-passage percolation model has several interpretations. It can be related to the Totally Asymmetric Simple Exclusion Process (TASEP) and to local random growth models. It is a basic example of a solvable model in the KPZ universality class. It has been less clear to what extent the two-time problem is also solvable but recently there has been some developments in this direction [1, 5, 9, 13, 17, 18]. The approach in this paper is different in many ways from that in our previous work [17]. It is closer to standard computations for determinantal processes, more straightforward and simpler.
To define the model, let \(\left( w(i,j)\right) _{i,j\ge 1}\) be independent geometric random variables with parameter q,
Consider the last-passage times
where the maximum is over all up/right paths from (1, 1) to (m, n), see [14]. We are interested in the correlation between \(G(m_1,n_1)\) and \(G(m_2,n_2)\), when \((m_1,n_1)\) and \((m_2,n_2)\) are ordered in the time-like direction, i.e. \(m_1<m_2\) and \(n_1<n_2\). To see why this is called a time-like direction, and give one reason why we are interested in the two-time problem, let us reinterpret the model as a discrete polynuclear growth model. It is clear from (1.1) that
Let \(G(m,n)=0\) if \((m,n)\notin \mathbb {Z}_+^2\), and define the height function h(x, t) by
for \(x+t\) odd, and extend it to all \(x\in \mathbb {R}\) by linear interpolation. Then (1.2) leads to a growth rule for h(x, t) and this is the discrete time and space polynuclear growth model. We think of \(x\mapsto h(x,t)\) as the height above x at time t, and we get a random one-dimensional interface. Let the constants \(c_i\) be given by (2.1). It is known, see [15], that the rescaled process
as a process in \(\eta \in \mathbb {R}\) for a fixed \(t>0\), converges as \(T\rightarrow \infty \) to \(\mathcal {A}_2(\eta )-\eta ^2\), where \(\mathcal {A}_2(\eta )\) is the Airy-2-process [21]. In particular, for any fixed \(\eta ,t\),
where \(F_2\) is the Tracy–Widom distribution, and
is the Airy kernel. The two-time problem is concerned with the question of the correlation between heights at different times. What is the limiting joint distribution of \(\mathcal {H}_T(\eta _1,t_1)\) and \(\mathcal {H}_T(\eta _2,t_2)\) for \(t_1<t_2\), as \(T\rightarrow \infty \)? From (1.3), we see that this is related to understanding the correlation between last-passage times in the time-like direction. That a time separation of order T is the correct order to get non-trivial correlations is quite clear if we think about how much random environment e.g. G(n, n) and G(N, N), \(n<N\), share. It can also be seen from the slow de-correlation phenomenon, see [4, 12]. Looking at (1.4) we see that we have the fluctuation exponent 1/3 (fluctuations have order \(T^{1/3}\)), the spatial correlation exponent 2/3, and we also have the time correlation exponent 1 = 3/3 as explained. This is the KPZ 1:2:3 scaling. For further references and more on random growth models in the KPZ-universality class and related interacting particle systems, we refer to the survey papers [2, 3, 22].
The main result of the present paper is a limit theorem for the following two-time probability. Fix m, M, n, N with \(1\le m<M\) and \(1\le n<N\). For \(a,A\in \mathbb {Z}\), we will consider the probability
in the appropriate scaling limit. The result is formulated in Theorem 2.1 below.
The first studies of the two-time problem, using a non-rigorous based on the replica method, was given by Dotsenko in [9, 10], see also [11]. However, the formulas are believed not to be correct [5]. The replica method has also been used by De Nardis and Le Doussal [5], to derive very interesting results in the limit \(t_1/t_2\rightarrow 1\) and, for arbitrary \(t_1/t_2\), in the partial tail of the joint law of \(\mathcal {H}_T(\eta _1,t_1)\) and \(\mathcal {H}_T(\eta _2,t_2)\) when \(\mathcal {H}_T(\eta _1,t_1)\) is large positive. In Le Doussal [18] gives a conjecturally exact formula for the limit \(t_1/t_2\rightarrow 0\). See also [13] for some rigorous work on this with quantitative results for the height correlation in the stationary case, which is not investigated here. We will not discuss these limits although to do so would be interesting. There are very interesting experimental and numerical results on the two-time problem by K. A. Takeuchi and collaborators, see [6, 23, 24].
Recently there has been a striking new development on the two-time problem, and more generally the multi-time problem, by Baik and Liu [1]. They consider the totally asymmetric simple exclusion process (TASEP) in a circular geometry, the periodic TASEP. Baik and Liu are able to give formulas for the multi-time distribution as contour integrals of Fredholm determinants, and take the scaling limit in the so-called relaxation time scale, \(T=O(L^{3/2})\), where L is the period. In principle their formulas include the problem studied here, but they are not able to take the scaling limit that we study in this paper. It would be interesting to understand the relation between the two approaches. For some comments on the multi-time problem in the setting used here see Remark 2.2. A related problem is to understand the Markovian time evolution of the whole limiting process with some fixed initial condition, the so called KPZ-fixed point. There has recently been very interesting progress on this problem by Matetski, Quastel and Remenik, see [19, 20].
An outline of the paper is as follows. In Sect. 2 we give the formula for the two-time distribution using an integral of a Fredholm determinant and state the main theorem. The main theorem is proved in Sect. 3 using a sequence of lemmas proved in Sects. 4 and 5. In Sect. 7, we briefly discuss the relation to the result in our previous work [17].
Notation Throughout the paper \(1(\cdot )\) denotes an indicator function, \(\gamma _r(a)\) is a positively oriented circle of radius r around the point a, and \(\gamma _r=\gamma _r(0)\). Also, \(\Gamma _c\) is the upward oriented straight line through the point c, \(t\mapsto c+it\), \(t\in \mathbb {R}\).
2 Results
Let \(0<t_1<t_2\), \(\eta _1,\eta _2\in \mathbb {R}\) and \(\xi _1,\xi _2\in \mathbb {R}\) be given. Furthermore T is a parameter that will tend to infinity. To formulate the scaling limit we need the constants,
We will investigate the asymptotics of the probability distribution defined by (1.5). The appropriate scaling is then
Let \(\Delta t=t_2-t_1\), and write
Introduce the notation
We will now define the limiting probability function. Before we can do that we need to define some functions. Fix \(\delta \) such that
and define
and
Using these, we can define the functions
Let u be a complex parameter and set
Consider the space
and define the following matrix kernel on X,
K(u) defines a trace-class operator on X, which we also denote by K(u). Let \(\gamma _r\) denote a circle around the origin of radius r with positive orientation. We define the two-time probability distribution by
where \(r>1\).
We can now formulate our main theorem.
Theorem 2.1
Let P(a, A) be defined as in (1.5) and consider the scaling (2.2). Then,
The theorem will be proved in Sect. 3. The fact that K(u) is a trace-class operator is Lemma 4.1 below.
The formula for the two-time distribution can be written in different ways. In Sect. 6, we will give formulas suitable for studying the limits \(\alpha \rightarrow 0\), \(\alpha \rightarrow \infty \) and expansions in \(\alpha \) and \(1/\alpha \) respectively. We will not discuss these expansions here, but refer to [7] for more on this and comparison with the results in [18].
For comments on the relation between this formula and the formula derived in [17], see the discussion in Sect. 7.
Remark 2.2
It would be interesting to be able to prove the same type of scaling limit for the multi-time case, i.e. to consider the probability function
where \(m_1< m_2<\dots < m_L\), and \(n_1< n_2<\dots < n_L\). It is possible to write a formula analogous to (3.17) below but with \(L-1\) contour integrals. This can be proved in a very similar way as the proof of (3.17). We hope to say more on this problem in future work.
3 Proof of the main theorem
In this section we will prove the main theorem. Along the way we will use several lemmas that will be proved in Sects. 4 and 5.
Write
for \(m\ge 0\), and a fixed \(N\ge 1\). Let \(\mathbf {G}(0)=0\). By \(\Delta \) we denote the finite difference operator defined on functions \(f:\mathbb {Z}\mapsto \mathbb {C}\) by \(\Delta f(x)=f(x+1)-f(x)\), which has the inverse
for all functions f for which the series converges. The negative binomial weight is
for \(m\ge 1\), \(x\in \mathbb {Z}\). Write
Note that \(\mathbf {G}(m)\in W_N\).
The following proposition is the starting point for the proof. It is proved in [16] following the paper by Warren [25], see also [8] for a more systematic treatment.
Proposition 3.1
The vectors \((\mathbf {G}(m))_{m\ge 0}\) form a Markov chain with transition function
for any \(\mathbf {x},\mathbf {y}\in W_N\), \(m>\ell \ge 0\).
Write
and
We can the write
Here we would like to perform the sum over \(\mathbf {y}\), which is straightforward, and then the sum over \(\mathbf {x}\), which is tricky since we cannot use the Cauchy–Binet identity directly. An important step is part a) of the following lemma, which is proved in Sect. 4. The proof of (3.7) uses successive summations by parts and generalizes the proof of Lemma 3.2 in [16].
Lemma 3.2
Let \(f,g:\mathbb {Z}\mapsto \mathbb {R}\) be given functions and assume that there is an \(L\in \mathbb {Z}\) such that \(f(x)=g(x)=0\) if \(x<L\).
-
(a)
Let \(a_i,d_i\in \mathbb {Z}\), \(1\le i\le N\) and fix k, \(1\le k\le N\). Then,
$$\begin{aligned}&\sum _{\mathbf {x}\in W_{N,k}(a)}\det \big (\Delta ^{j-a_i}f(x_j-y_i)\big )_{1\le i,j\le N}\det \big (\Delta ^{d_i-j}g(z_i-x_j)\big )_{1\le i,j\le N}\nonumber \\&\quad =\sum _{\mathbf {x}\in W_{N,k}(a)}\det \big (\Delta ^{k-a_i}f(x_j-y_i)\big )_{1\le i,j\le N}\det \big (\Delta ^{d_i-k}g(z_i-x_j)\big )_{1\le i,j\le N}. \end{aligned}$$(3.7) -
(b)
For \(1\le n\le N\), we have the identity
$$\begin{aligned} \sum _{\mathbf {x}\in W_{N,N}(A)}\det \big (\Delta ^{i-n}w_m(x_i-y_j)\big )_{1\le i,j\le N}=\det \big (\Delta ^{i-n-1}w_m(A-y_j)\big )_{1\le i,j\le N}.\nonumber \\ \end{aligned}$$(3.8)
If we use (3.7) and (3.8) in (3.6), we find
Before we show how we can use the Cauchy–Binet identity to do the summation in (3.9), we will modify it somewhat. Below, this modification will be a kind of orthogonalization procedure, and will be important for obtaining a Fredholm determinant. Let \(A=(a_{ij})\) and \(B=(b_{ij})\) be two \(N\times N\)-matrices that satisfy \(a_{ij}=0\) if \(j>i\) and \(b_{ij}=0\) if \(j<i\), so that A is lower- and B upper-triangular. Assume that
For \(x\in \mathbb {Z}\), \(1\le i,j\le N\), we define
and
where \(w_m\) is the negative binomial weight (3.2). If we shift \(x_i\rightarrow x_i+a\), \(1\le i\le N\), in (3.9), and use (3.10), (3.11) and (3.12), we get
This formula is the basis for the next lemma, the proof of which is based on the Cauchy–Binet identity. However, because of the restriction \(x_n< 0\) in the summation in (3.13), we cannot apply the identity directly. In order to state the result we need some further notation. Define
Let u be a complex parameter and set
Lemma 3.3
We have the formula,
for any \(r>1\).
The lemma is proved in Sect. 4. The contour integral come from the need to capture the restriction \(x_n<0\) and still use the Cauchy–Binet identity.
We now come to the choice of the matrices A and B. The aim is to get a good formula for \(f_{0,1}\) and \(f_{1,2}\) and make it possible to write the determinant in (3.17) as a Fredholm determinant suitable for asymptotic analysis. Define
Using a generating function for the negative binomial weight (3.2), it is straightforward to show that for all \(m\ge 1\), \(k,x\in \mathbb {Z}\),
if \(r>1\). For \(k,x\in \mathbb {Z}\), \(m\ge 1\), \(\epsilon \in \{0,1\}\) and \(0<\tau <1\), we define
Note that \(\beta _0^{\epsilon }=1\) and \(\beta _k^{\epsilon }=0\) if \(k\ge 1\). By expanding \((z-\zeta )^{-1}\) in powers of \(\zeta /z\), we see that
provided \(|z|>\tau \).
We now define the matrices A and B. Let c(i) be a conjugation factor defined below in (3.25) which we need to make the asymptotic analysis work. Set
From the properties of \(\beta _k^\epsilon \), we see that \((a_{ik})\) is lower- and \((b_{kj})\) upper-triangular, and that the condition (3.10) is satisfied.
Lemma 3.4
If \(f_{0,1}\) and \(f_{1,2}\) are defined by (3.11) and (3.12) respectively, and \(a_{ik}\) and \(b_{kj}\) by (3.22), then
where \(0<\tau<1<r\).
The proof of the lemma, which will be given in Sect. 4, is a straightforward computation using the definitions and (3.21).
We now turn to rewriting the determinant in (3.17) as a Fredholm determinant and performing the asymptotic analysis. The conjugation factor c(i) in (3.22) is given by
where \(\delta >0\) is fixed, and satisfies (2.5), and \(c_1\) is given by (2.1). Let \(\tau _1, \tau _2,\rho _1,\rho _2\) and \(\rho _3\) be radii such that
We denote by \(\gamma _\rho (1)\) a positively oriented circle around the point 1 with radius \(\rho \). For \(\epsilon \in \{0,1\}\) and \(1\le i,j\le N\), we define
and
We also define, for \(\epsilon \in \{0,1\}\) and \(1\le i,j\le N\),
compare with (2.10) and (2.11).
We can now express \(L_p\), \(p=1,2\), in terms of these objects.
Lemma 3.5
We have the formulas
and
The proof is based on (3.14), (3.15), and Lemma 3.4, and suitable contour deformations in order to get the contours into positions that can be used in the asymptotic analysis, see Sect. 4.
Combining (3.16) with Lemma 3.5 we obtain
where
and we also set \(M_{u}(i,j)=0\) if \(i,j\notin \{1,\dots ,N\}\). Thus we have the formula
Next, we want to rewrite the determinant in (3.37) in a block determinant form, corresponding to \(i\le n\) and \(i>n\), and similarly for j. For \(r,s\in \{1,2\}\), and \(x,y\in \mathbb {R}\), we define
where \([\cdot ]\) denotes the integer part. The right side of (3.38) does not depend on r or s explicitely but we have \(x<0\) for \(r=1\) and \(x\ge 0\) for \(r=2\), and correspondingly for y depending on s. Let \(\Lambda =\{1,2\}\times \mathbb {R}\) and define the measures
On \(\Lambda \) we define a measure \(\rho \) by
for every integrable function \(f:\Lambda \mapsto \mathbb {R}\). \(F_{u}\) defines an integral operator \(F_{u}\) on \(L^2(\Lambda ,\rho )\) with kernel \(F_{u}(r,x;s,y)\). Note that the space \(L^2(\Lambda ,\rho )\) is isomorphic to the space X defined in (2.13), and we can also think of \(F_{u}\) as a matrix operator.
Lemma 3.6
We have the identity,
This is straightforward, using Fredholm expansions, and the lemma will be proved in Sect. 4.
We can now insert the formula (3.40) into (3.37). This leads to a formula that can be used for taking a limit, but before considering the limit, we have to introduce the appropriate scalings. For \(s=1,2\), we define
where \(c_0\) is given by (2.1). The next lemma follows from (3.37), Lemma 3.6, and (3.41), see Sect. 4.
Lemma 3.7
We have the formula,
Theorem 2.1 now follows by combining this lemma with the next lemma which will be proved in Sect. 5.
Lemma 3.8
Consider the scaling (2.2) and let K(u) be the matrix kernel defined by (2.14). Then,
uniformly for u in a compact set.
4 Proof of Lemmas
In this section we will prove the lemmas that were used in Sect. 3. Some results related to the asymptotic analysis will be proved in Sect. 5.
Proof of Lemma 3.2
Write
so that
Hence, it is enough to prove the statement with \(W_{N,k}(a)\) replaced by \(W^*_{N,k}(t)\). Let \(a_i, b_i, c_i, d_i\in \mathbb {Z}\), \(1\le i,j\le N\), and let \(k<\ell \le N\). Assume that \(b_{\ell -1}=b_\ell -1\), and \(c_{\ell }=c_{\ell +1}\) if \(\ell <N\). Set
Then,
To prove (4.1), we use the summation by parts identity,
Consider the \(x_\ell \)-summation in the left side of (4.1) with all the other variables fixed. Let \(x_{\ell +1}=\infty \) if \(\ell =N\) and let \(\Delta _x\) denote the finite difference with respect to the variable x. Using (4.2) in the second inequality we get
If \(\ell =N\), then the first boundary term in (4.3) is \(=0\). This follows since \(\Delta ^{d_i-c_\ell }g(z_i-\infty )=0\) (assumption that all series are convergents, expressions well-defined), so one column in the second determinant the first boundary term in (4.3) is \(=0\). If \(\ell <N\), then the first boundary term in (4.3) is \(=0\) because \(c_{\ell }=c_{\ell +1}\), and \(x_\ell \rightarrow x_{\ell +1}\) means that columns \(\ell \) and \(\ell +1\) will be identical in the second determinant. Since \(b'_{\ell }=b_\ell -1=b_{\ell -1}\), we see that columns \(\ell \) and \(\ell -1\) in the first determinant in the second boundary term in (4.3) will be identical.
Similarly, if \(1\le \ell <k\), and \(c_{\ell +1}=c_\ell +1\), \(b_\ell =b_{\ell -1}\), then
where
The proof of (4.4) is analogous to the proof of (4.1).
To prove Lemma 3.2, we apply (4.1) successively to \(x_N,x_{N-1},\dots ,x_{k+1}\), and then to \(x_N,x_{N-1},\dots ,x_{k+2}\) etc., and then finally just to \(x_N\). Similarly, we apply (4.4) to \(x_1,x_2,\dots , x_{k-1}\), then to \(x_1,x_2,\dots , x_{k-2}\), and finally just to \(x_1\). This proofs part (a) of the lemma.
Part (b) of the lemma follows from the identity
To prove (4.5), first sum over \(x_N\) from \(x_{N-1}\) to a in the last row. This gives \(\Delta ^{N-1-n}f_j(a+1)-\Delta ^{N-1}f_j(x_{N-1})\). The last term does not contribute since it is the same as in row \(N-1\). We can now sum over \(x_{N-1}\) from \(x_{N-2}\) to a in row \(N-1\) etc. In this way we obtain (4.5). \(\square \)
Proof of Lemma 3.3
We see that
Now, for any \(r>0\),
Summing over \(\ell \ge n\) and assuming that \(r>1\), we get
Since,
it follows from (4.6), (4.7), and the Cauchy–Binet identity that
\(\square \)
Proof of Lemma 3.4
It follows from (3.11), (3.21), and (3.22), that
Similarly, by (3.12), (3.21) and (3.22),
This proves the lemma. \(\square \)
Proof of Lemma 3.5
Recall the condition (3.26) and choose \(r_1, r_2\) so that \(r_1>r_2>1+\max (\rho _1,\rho _2)\), which means that \(\gamma _{r_i}(1)\) surrounds \(\gamma _{\rho _i}\) and \(\gamma _{\tau _i}\), \(i=1,2\). It follows from (3.23) and (3.24), that
Since \(r_1>r_2\),
and we obtain
We now deform \(\gamma _{r_2}(1)\) to \(\gamma _{\rho _2}(1)\). Doing so, we cross the pole at \(w=\omega \), and hence
In \(I_1\) we can shrink \(\gamma _{r_1}(1)\) to \(\gamma _{\rho _1}(1)\). We then cross the pole at \(z=\zeta \) (but not \(z=w\) since \(\rho _2<\rho _1\)). Thus, by (3.27),
We note that
since \(|w|>|\zeta |\), and hence by (3.29),
Also
and we obtain
Deform \(\gamma _{r_1}(1)\) to \(\gamma _{\rho _1}(1)\). We then cross the pole at \(z=\zeta \) and we obtain, using (3.30),
Combining (4.8), (4.9), (4.11), (4.13) and (3.31), we get (3.33).
Consider next,
where now \(r_2>r_3>1+\max (\rho _1,\rho _2)\). Thus,
and consequently
We now deform \(\gamma _{r_3}(1)\) to \(\gamma _{\rho _3}(1)\), and doing so we pass the pole at \(z=\zeta \), and find
In \(J_1\) we deform \(\gamma _{r_2}(1)\) to \(\gamma _{\rho _2}(1)\). Since \(\rho _2>\rho _3\), we only cross the pole at \(w=\omega \), and we get
Using (4.10), we find
which gives (3.34) and the lemma is proved. \(\square \)
Proof of Lemma 3.6
We start with the right side of (3.40),
where we recall that \(M_{u}(i,j)=0\) if \(i,j\notin \{1,\dots ,N\}\). \(\square \)
Proof of Lemma 3.7
By the formula (3.37) for P(a; A) and Lemma 3.6, we see that
We have the Fredholm expansion,
The change of variables \(x_p\rightarrow c_0(t_1T)^{1/3}x_p\) gives
Take the factor \(c_0(t_1T)^{1/3}\) into row p. We see then that the right side of (4.15) equals,
Combining this with (4.14) we have proved the lemma. \(\square \)
We want to prove that the operator K(u) in the definition of the two-time distribution is a trace-class operator.
Lemma 4.1
The operator K(u) defined by (2.14) is a trace-class operator on the space X given by (2.13).
Proof
Write
so that
By splitting K(u) into several parts and factoring out multiplicative constants, we see that it is enough to prove that
is a trace-class operator on X for \(A=S_1, T_1, S_2^*,S_3^*\). We can think of A as an operator on \(L^2(\Lambda ,\rho )\) instead, where \(\Lambda =\{1,2\}\times \mathbb {R}\) and \(\rho \) is given by (3.39).
Define the kernels
Using the definitions, we see that
To get kernels on \(L^2(\Lambda ,\rho )\), we define
for \(r_1=1,2\), and
for \(r_1=1,2\). Furthermore, we define
so \(S_1=a_1a_2\). Similarly, we see that \(T_1=\tilde{a}_1\tilde{a}_2\), \(S_2^*=b_1b_2\) and \(S_3^*=c_1c_2\). Using (2.5) and asymptotic properties of the Airyfunction, we see that \(a_1,a_2,b_1,b_2, c_1, c_2\) are square integrable over \(\mathbb {R}^2\), and also over \(\mathbb {R}\) if we fix one of the variables to be zero. It follows from this that \(a_1, a_2, \tilde{a}_1,\dots , c_2\) are Hilbert-Schmidt operators on \(L^2(\Lambda ,\rho )\). Since the composition of two Hilbert-Schmidt operators is a trace-class operator, we have that \(S_1, T_1, S_2^*\) and \(S_3^{*}\) are trace-class operators on \(L^2(\Lambda ,\rho )\), and hence K(u) is a trace-class operator also. \(\square \)
5 Asymptotic analysis
In this section we will prove Lemma 3.8. The proof has several steps and we will split it into a sequence of lemmas. The proofs of these lemmas will appear later in the section.
For \(k=1,2,3\), we define the rescaled kernels
Lemma 5.1
Uniformly, for x, y in a compact subset of \(\mathbb {R}\), we have the limits
and
The lemma is proved below. In order to prove the convergence of the Fredholm determinant we also need some estimates.
Lemma 5.2
Assume that \(|\xi |,|\eta |\le L\) for some fixed L. If we choose \(\delta \) in (3.25) sufficiently large, depending on q and L, there are positive constants \(C_0, C_1, C_2\) that only depend on q and L, so that for all x, y satisfying
we have the estimates
Here \((x)_+=\max (0,x)\).
The proof is given below. We now have the estimates that we need to prove Lemma 3.8
Proof of Lemma 3.8
Recall from (2.12) and (2.14) that
\(s=1,2\). It follows from Lemma 5.1 that
for \(r,s\in \{1,2\}\), uniformly for u, x, y in compact sets. From (5.5) we see that for all \(\xi ,\eta ,u\) in compact sets there are positive constants \(C_0,C_1\) so that
for \(r,s\in \{1,2\}\) and all \(x,y\in \mathbb {R}\). Note that, by definition \(\tilde{F}_{u,T}\) is zero if x, y do not satisfy (5.4). We can expand the Fredholm determinant,
in its Fredholm expansion. It follows from (5.6), (5.7) and Hadamard’s inequality that we can take the limit \(T\rightarrow \infty \) in (4.15) and get
This completes the proof. \(\square \)
Consider
with the scalings (\(K\rightarrow \infty \), \(\eta ,\xi ,v\) fixed),
Here the constants \(c_i\) are given by (2.1). Write
If \(\eta =\xi =v=0\), then f(w) has a double critical point at
Define
The local asymptotics around the critical point is given by the next lemma.
Lemma 5.3
Fix \(L>0\) and assume that \(|\xi |,|\eta |, |v|\le L\). Furthermore, assume that we have the scaling (5.9). Then, uniformly for \(w'\) in a compact set in \(\mathbb {C}\)
where
Proof
Let
so that
Then \(f_1'(w)\) has a double zero at \(w_c\) only if the constant \(c_2=2\sqrt{q}/(1-\sqrt{q})\). A computation gives
and we find
Also,
and
Using (5.16), (5.17) and (5.18) in (5.15), we obtain
as \(K\rightarrow \infty \). \(\square \)
To prove the estimates that we need, we use some explicit contours in (3.27) to (3.30). Let \(d>0\) and define
and
for \(|\sigma |\le \pi K^{1/3}\), where K is as in (5.9). Thus, \(w_1\) gives a circle around the origin of radius \(w_c(1-\frac{d}{K^{1/3}})\), and \(w_2\) gives a circle of radius \( \sqrt{q}(1-\frac{d}{K^{1/3}})\) around 1.
Lemma 5.4
Fix \(L>0\). Assume that we have the scaling (5.9) and that \(|\xi |,|\eta |, |v|\le L\). Then, there are positive constants \(C_j\), \(1\le j\le 4\) that only depend on q and L, so that if \(C_1\le d\le C_2\), then
and
for \(|\sigma |\le \pi K^{1/3}\).
We will also need estimates that work for large v.
Lemma 5.5
Assume that \(|\xi |,|\eta |\le L\) for some fixed \(L>0\), and assume that we have the scaling (5.9) and v is such that \(k\ge 0\). Then, we can choose \(d=d(v)\ge C_0\), so that
for \(|\sigma |\le \pi K^{1/3}\), where \(C_0,C_1,C_2,\mu _1,\mu _2\) are positive constants that only depend on q and L. Similarly, there is a choice of \(d=d(v)\) so that
These two Lemmas will be proved below. We can use Lemma 5.3 and Lemma 5.4 to prove Lemma 5.1.
Proof of Lemma 5.1
It follows from (3.25), (3.27) and (5.12) that
Let \(\Gamma _D\) denote the vertical line through D oriented upwards, \(\mathbb {R}\ni t\mapsto D+\mathrm {i}t\). Let \(D_1>D_2>0\), \(d_1,d_2>0\) be such that
\(r=1,2\), where \(C_1,C_2\) are the constants in Lemma 5.4 with some fixed L arbitrarily large. We choose the following parametrizations in (5.25),
where \(K=K_1=(t_1T)^{1/3}\) in (5.19), (5.19), and
where \(K=K_2=(\Delta tT)^{1/3}\),
Recall the condition (3.26) on the radii. Let
Now, a computation shows that, for some constant C,
for all \(\sigma _i\) satisfying (5.28). Thus, for x, y in a compact set, we have the following bound on the integrand in (5.25),
where the last inequality follows from Lemma 5.4.
For \(\sigma _i\) in a bounded set, we see that
It follows from (2.2) that
and hence
Write \(z'=-\mathrm {i}\sigma _1+D_1\), \(w'=-\mathrm {i}\sigma _2+D_2\), \(\zeta '=\mathrm {i}\sigma _3+d_1\), \(\omega '=\mathrm {i}\sigma _4+d_2\). Note that
It follows from Lemma 5.3, (5.25), (5.32), (5.31) and the dominated convergence theorem that
and we have the condition
Define
and let
If \(d,D>0\), we have the formulas,
with absolutely convergent integrals. Using (5.37), we see that
It follows from these formulas, (5.39) and (5.40) that \(S_1\) is also given by (2.6).
The proof of (5.3) is identical with \(D_1\) replaced by \(D_3\) satisfying (5.37). The integral formula for \(T_1\) reads
The other cases are treated similarly. For \(S_2\) and \(S_3\) we get the formulas
and
This proves Lemma 5.1. \(\square \)
Proof of Lemma 5.2
Consider first \(\tilde{A}_{1,T}\). By Lemma (5.4), we can choose \(d_1\) and \(d_2\), with \(d_1<\alpha d_2\), so that
where \(C_3,C_4\) are some positive constants independent of \(\sigma _1\) and \(\sigma _2\). By Lemma 5.5, we can choose \(d=d_3(x)\ge C_0\), and \(d=d_4(y)\ge C_0\), so that
It is not difficult to check that if \(z=w_2(\sigma _1,d_1)\), \(w=w_2(\sigma _2,d_2)\), \(\zeta =w_1(\sigma _3,d_3(x))\) and \(\omega =w_1(\sigma _4,d_4(y))\), then there is a constant \(C_5\) so that
and
Introducing these parametrizations into (5.25) and using the estimates above, we find
We see that for large enough |x|, we can choose \(\delta \) so large that
for some positive constants \(C_1,C_2\). This proves the estimate for \(\tilde{A}_{1,T}\). The proof for \(\tilde{B}_{1,T}\) is completely analogous.
Consider now \(\tilde{A}_{3,T}\),
Using Lemma 5.5, we see that, just as for \(\tilde{A}_{1,T}\), we can choose \(d_1(y)\) and \(d_2(x)\) so that
and we get the estimate
This gives us the estimate we want by choosing \(\delta \) large enough. The proof for \(\tilde{A}_{2,T}\) is analogous. \(\square \)
The statements in Lemma 5.4 and in Lemma 5.5 are consequences of two other lemmas that we will now state and prove. The first lemma is concerned with the decay along the paths given by \(w_1(\sigma )\) and \(w_2(\sigma )\).
Lemma 5.6
Assume that we have the scaling (5.9) and let \(|\xi |,|\eta |\le L\) for some fixed \(L>0\). There are positive constants \(C_1,C_2,C_3,C_4\) that only depend on q and L, so that if
then for \(|\sigma |\le \pi K^{1/3}\),
for all \(v\in \mathbb {R}\). Furthermore, for \(|\sigma |\le \pi K^{1/3}\),
for all \(v\le 0\) such that \(k\ge 0\), and all v such that \(|v|\le L\).
Proof
Recall the definition of f(w) in (5.10) and the parametrizations (5.19) and (5.20). Define
\(r=1,2\), \(|\sigma |\le \pi K^{1/3}\). Note that for any real numbers \(\alpha ,\beta \),
Let \(\beta =K^{-1/3}\), \(\alpha _1=w_c(1-dK^{-1/3})\), \(\alpha _2=\alpha _1/(1-q)\). Then a computation using (5.51) and (5.52) gives
By symmetry it is enough to consider \(0\le \sigma \le \pi K^{1/3}\). We have to compute
Now,
and using (5.9) a computation gives
Since \(|\xi |,|\eta |\le L\), we see that
where
We note that we can choose \(C_1\) and \(C_2\), depending only on q and L, so that if \(C_1\le d\le C_2K^{1/3}\), then \(\Delta _1\ge 0\) and \(\Delta _2\ge 0\), and also
Thus, we see from (5.54) and (5.55) that
provided that \(C_1\le d\le C_2K^{1/3}\). Consequently, by (5.53),
since
It follows, by integration, that, for \(0\le \sigma \le \pi K^{1/3}\),
since by convexity \(\sin t\ge 2t/\pi \) for \(0\le t\le \pi /2\). This proves the estimate (5.49).
Next, we turn to the proof of (5.50) which is similar. In this case we get
where \(\beta =K^{-1/3}\). Let \(\alpha _1=\sqrt{q}(1-d\beta )\), \(\alpha _2=\frac{1}{q} \alpha \). Then, using (5.52), we obtain
Now,
and a computation gives
where
If \(|\xi |,|\eta |,|v|\le L\), we see that we can choose \(C_1, C_2\), depending only on q, L, so that if \(C_1\le d\le C_2K^{1/3}\), the \(\Delta \ge 0\), and we obtain
If \(|\xi |,|\eta |\le L\) and \(v\le 0\), we can also choose \(C_1, C_2\) so that \(\Delta \ge 0\) if \(C_1\le d\le C_2K^{1/3}\). Also, we see that
if \(v\le 0\) or \(|v|\le L\). Assume that \(C_2\) is such that \(\alpha _1\ge \sqrt{q}/2\). Then (5.58), (5.60) and (5.61) give
if we choose \(C_1\) so that
for \(d\ge C_1\). Since \(\alpha _1\le \sqrt{q}\), \(\alpha _1\le 1/\sqrt{q}\),
and (5.57) gives
We can now proceed, as for \(g_1\), to prove that
This completes the proof of the Lemma. \(\square \)
The next Lemma is concerned with the decay for large |v|.
Lemma 5.7
Assume that we have the scaling (5.9) and that v is such that \(k\ge 0\), which will always be the case. Also, assume that \(|\xi |,|\eta |\le L\) for some \(L>0\). There are positive constants \(\mu _1,\mu _2,\mu _3\) that only depend on q, L, and a choice \(d=d(v)\) satisfying (5.48) so that
There is also a choice \(d=d(v)\) satisfying (5.48) so that
If we assume that \(|v|\le L\), we can choose d independent of v in some interval so that (5.62) and (5.63) hold.
Proof
Using (5.51) we see that
so we want to estimate \(g_1(0)-\log f(w_c)\) from below, and then make a good choice of d. We see that
To estimate this expression, we will use the inequalities
for \(1/2\le x\le 1\), and
for \(x\ge 0\). It follows from (5.64) and these inequalities that
Substitute the expressions in (5.9). After some manipulation this gives
If \(|v|\le L\), we see that if we choose d so that \(C_1'\le d\le C_2'\), then
Here \(C_1',C_2', C_3'\) only depend on q, L. If \(v\le 0\), then it follows from (5.67) that
Choose \(d=\epsilon \sqrt{-v}\). Then, by (5.68),
Choose \(D_1\) large, depending on only q, L, so that
if \(\sqrt{-v}\ge D_1\). Since \(k\ge 0\), there is a constant \(D_2\) so that \(\sqrt{-v}\le D_2K^{1/3}\). The condition (5.48) becomes
which is satisfied if
We can choose \(D_1\) so large that \(C_1/D_1\) is as small as we want, and hence we can choose \(\epsilon \) so small that
It then follows from (5.69) that
for \(\sqrt{-v}\ge D_1\). By adjusting \(\mu _3\), we see that (5.62) holds if \(v\le 0\).
If \(v\ge 0\), we choose a d satistying (5.48) depending on q, L, but not on v or K. It follows from (5.68) that there are constants \(\mu _1\) and \(\mu _3'\), so that
Hence (5.62) holds also when \(v\ge 0\).
To prove (5.63) we consider instead
by (5.65) and (5.66). Into this estimate we insert the expressions in (5.9), and after some computation we get
We can now proceed in analogy with the previous case to show (5.63). \(\square \)
6 More formulas for the two-time distribution
In this section we give an alternative formula for the two-time distribution, see Proposition 6.1 below.
Recall the notation (5.38),
Looking at (5.40), we see that it is natural to write
since we then get the formulas
for any \(d,D>0\). We can think of (6.2) as the kernel of an integral operator on \(L^2(\mathbb {R}_+)\).
In order to give a different formula for the two-time distribution, we need to define several kernels. We will write
Let
and
We will also need the following kernels. Let
Define
and
The kernels \(M_i\) and \(k_i\) depend on the parameters \(\alpha , \xi _1, \Delta \xi , \eta _1, \Delta \eta \) and \(\delta \). When we need to indicate this dependence we write \(M_i(\alpha , \xi _1, \Delta \xi , \eta _1, \Delta \eta , \delta )\) and \(k_i(\alpha , \xi _1, \Delta \xi , \eta _1, \Delta \eta , \delta )\). We then think of \(\xi _2\) and \(\eta _2\) as functions of \(\alpha \), \(\xi _1\) and \(\Delta \xi \), and \(\alpha \), \(\eta _1\) and \(\Delta \eta \) respectively. Explicitly,
Let
On Y, we define a matrix operator kernel Q(u) by
where
We will write \(Q(u,\alpha , \xi _1, \Delta \xi , \eta _1, \Delta \eta , \delta )\) to indicate the dependence on all parameters.
Proposition 6.1
The two-time distribution (2.15) is given by
where \(r>1\).
We will give the proof below. The formula (6.22) is suitable for investigating the limit \(\alpha \rightarrow 0\) (long time separation). For more on this limit see [7]. To study the limit \(\alpha \rightarrow \infty \) (short time separation), we can use (6.22) and the next Proposition which gives an \(\alpha \) and \(1/\alpha \) relation. Let
To indicate the dependence of the kernel K(u) on all parameters we write \(K(u,\alpha , \xi _1, \Delta \xi , \eta _1, \Delta \eta , \delta )\).
Proposition 6.2
We have the formula
where \(r>1\).
The proof is given below. Recall that
Combining the two Propositions above we see that
Note that \(\alpha \) is replaced by \(\beta =1/\alpha \), \(\xi _1\) and \(\Delta \xi \), as well as \(\eta _1\) and \(\Delta \eta \), are interchanged, and u is replaced by \(u^{-1}\). This formula is suitable for studying the limit \(\alpha \rightarrow \infty \) since this corresponds to \(\beta \rightarrow 0\), see [7]. Note that combining (6.17), (6.18) and (6.25), we get
We now turn to the proofs of the Propositions.
Proof of Proposition 6.1
Define the kernels
The factors involving \(\delta v\) have been introduced in order to get well-defined operators. We also define
From (5.39) and (5.41), we see that
by moving the z-integration contour. We then pick up a contribution from the pole at \(z=\alpha w\), which gives \(S_4\). It follows from (5.41), (5.42), (5.43), (6.29) and (6.28) that
From the definition of R(u), (6.30) and (6.31), we see that
Let \(p_i^{\pm }\) be the operator from \(L^2(\mathbb {R}_+)\) to \(L^2(\mathbb {R}_\pm )\) with kernel \(p_i(x,v)\), and \(q_i^{\pm }\) be the operator from \(L^2(\mathbb {R}_\pm )\) to \(L^2(\mathbb {R}_+)\) with kernel \(q_i(v,y)\). From the definition of K(u) and (6.32) it follows that
where
and
are matrix operators \(p:Y\mapsto X\) and \(q:X\mapsto Y\). Note that \(p_2^-=q_2^+=0\). Let
which gives an operator from Y to itself. A straightforward computation using (6.28), (6.10)–(6.16) and (6.6)–(6.8) shows that
From this we see that Q(u) is given by (6.21). In these computations we use (6.17) and (6.18) to get \(\xi _2,\eta _2\) from \(\xi _1,\Delta \xi , \eta _1,\Delta \eta \). The Proposition now follows from
\(\square \)
Proof of Proposition 6.2
To indicate the dependence of S, T and R(u) on all parameters we write \(S(\alpha ,\xi _1,\Delta \xi ,\eta _1,\Delta \eta ,\delta )\) etc. It is straightforward to check from the definitions that
and
It follows that
If we write
we see that
Let \(K^*_{\alpha }(u)(x,y)=\alpha ^{-1}K(\alpha ^{-1}y,\alpha ^{-1}x)\), and define \(V:X\mapsto X\) by
Note that \(V^2=I\). Since taking the adjoint and rescaling the kernel does not change the Fredholm determinant, we see that
Using these definitions a computation shows that
This operator has the same determinant as
Thus,
since the Fredholm determinant is independent of the value of \(\delta \) as long as the condition (2.5) is satisfied. Note that this condition is \(\delta >\max (\eta _1,\alpha \Delta \eta )\) so \(\beta \delta >\max (\Delta \eta ,\beta \eta _1)\) and we can replace \(\beta \delta \) with \(\delta \) as long as \(\delta >\max (\Delta \eta ,\beta \eta _1)\). \(\square \)
7 Relation to the previous two-time formula
The approach in the present paper can be modified to study the probability
under the same scaling (2.2).
Let
and modify the definition of S and T into
Define the matrix kernel
where R(u) is defined as in (2.12) but with S and T given by (7.2) and (7.3) instead. Then, under (2.2),
for any \(r>0\). From this formula, it is possible to derive the formula for the two-time distribution given in [17]. It should be possible to get the formula in [17] also by taking the partial derivative with respect to \(\xi _1\) in (2.15). We have not been able to carry out that computation.
References
Baik, J., Liu, Z.: Multi-point distribution of periodic TASEP. arXiv:1710.03284
Borodin, A., Petrov, L.: Integrable probability: from representation theory to MacDonald processes. Probab. Surv. 11, 1–58 (2014)
Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1(1), 1130001 (2012)
Corwin, I., Ferrari, P.L., Péché, S.: Universality of slow de-correlation in KPZ growth. Ann. Inst. Henri Poincar Probab. Stat. 48, 134–150 (2012)
De Nardis, J., Le Doussal, P.: Tail of the two-time height distribution for KPZ growth in one dimension. J. Stat. Mech. Theory Exp. 5, 053212 (2017)
De Nardis, J., Le Doussal, P., Takeuchi, K.A.: Memory and universality in interface growth. Phys. Rev. Lett. 118, 125701 (2017)
De Nardis, J., Johansson, K., Le Doussal, P. (in preparation)
Dieker, A.B., Warren, J.: Determinantal transition kernels for some interacting particles on the line. Ann. Inst. Henri Poincare Probab. Stat. 44(6), 1162–1172 (2008)
Dotsenko, V.: Two-time free energy distribution function in \((1+1)\) directed polymers. J. Stat. Mech. Theory Exp. 6, P06017 (2013)
Dotsenko, V.: Two-point free energy distribution function in (1+1) directed polymers. J. Phys. A 46(35), 355001 (2013)
Dotsenko, V.: On two-time distribution functions in (1+1) random directed polymers. J. Phys. A 49(27), 27LT01 (2016)
Ferrari, P.L.: Slow decorrelations in KPZ growth. J. Stat. Mech. Theory Exp. 7, P07022 (2008)
Ferrari, P.L., Spohn, H.: On time correlations for KPZ growth in one dimension. SIGMA Symmetry Integr. Geom. Methods Appl. 12(074), 23 (2016)
Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)
Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003)
Johansson, K.: A multi-dimensional Markov chain and the Meixner ensemble. Ark. Mat. 48, 437–476 (2010)
Johansson, K.: Two time distribution function in Brownian directed percolation. Commun. Math. Phys. 351, 441–492 (2017)
Le Doussal, P.: Maximum of an Airy process plus Brownian motion and memory in KPZ growth. Phys. Rev. E 96, 060101 (2017)
Matetski, K., Quastel, J., Remenik, D.: The KPZ fixed point. arXiv:1701.00018
Matetski, K., Quastel, J.: From the totally asymmetric simple exclusion process to the KPZ fixed point. arXiv:1710.02635
Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002)
Quastel, J.: Introduction to KPZ. Current Developments in Mathematics (2011), pp. 125–194. International Press, Somerville (2012)
Takeuchi, K.A.: Statistics of circular interface fluctuations in an off-lattice Eden model. J. Stat. Mech. 5, P05007 (2012)
Takeuchi, K.A., Sano, M.: Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence. J. Stat. Phys. 147, 853–890 (2012)
Warren, J.: Dyson’s Brownian motions, intertwining and interlacing. Electron. J. Probab. 12(19), 573–590 (2007)
Acknowledgements
I thank Jinho Baik for an interesting discussion and correspondence. Also, thanks to Mustazee Rahman for helpful comments on the paper.
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Johansson, K. The two-time distribution in geometric last-passage percolation. Probab. Theory Relat. Fields 175, 849–895 (2019). https://doi.org/10.1007/s00440-019-00901-9
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DOI: https://doi.org/10.1007/s00440-019-00901-9