Remarks on Superconcentration and Gamma Calculus: Applications to Spin Glasses

Abstract

This note is concerned with the so-called superconcentration phenomenon. It shows that the Bakry-Émery’s Gamma calculus can provide relevant bound on the variance of function satisfying a inverse, integrated, curvature criterion. As an illustration, we present some variance bounds for the Free Energy in different models from Spin Glasses Theory.

10.1 Introduction

Superconcentration phenomenon has been introduced by Chatterjee in [7] and has given birth to a lot a work (cf. [15] for a survey). Each of these works, used various ad-hoc methods to improve upon sub-optimal bounds given by classical concentration of measure (cf. [4, 10]). In this note, we want to show that the celebrated Gamma calculus from Bakry and Émery’s Theory is relevant to such improvements. To this task, we introduce an inverse, integrated, Γ₂ criterion which provides a useful bound on the variance of a particular function. As far as we know, this criterion seems to be new. We give below a sample of our modest achievement.

Denote by γ _n the standard Gaussian measure on $\mathbb {R}^n$ and by (P _t)_t≥0 the standard Ornstein–Uhlenbeck semigroup. Γ will stand for the so-called “carré du champ” operator, associated to the infinitesimal generator L =  Δ − x ⋅∇ of (P _t)_t≥0, and Γ₂ its iterated operator. We refer to Sect. 10.2 for more details about this topic.

Theorem 10.1.1

Let $f\,:\,\mathbb {R}^n\to \mathbb {R}$ be a regular function and assume that there exists $\psi \,:\,\mathbb {R}_+\to \mathbb {R}$ such that

• (1) for any t ≥ 0,

  $$\displaystyle \begin{aligned} \int_{\mathbb{R}^n}\Gamma_2(P_t f)d\gamma_n\leq \int_{\mathbb{R}^n}\Gamma(P_t f)d\gamma_n+\psi(t), \end{aligned} $$

  (10.1)
• (2)
  $$\displaystyle \begin{aligned} \int_0^\infty e^{-2t}\int_t^\infty e^{2 s}\psi(s)ds dt<\infty. \end{aligned}$$

Then the following holds

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_n}(f)\leq \bigg|\int_{\mathbb{R}^n}\nabla fd\gamma_n\bigg|{}^2+4\int_0^\infty e^{-2 t}\int_t^\infty e^{2s}\psi(s)dsdt. \end{aligned}$$

with |⋅| the standard Euclidean norm.

Remark

Equation (10.14) can be seen as an inverse, integrated, curvature inequality for the function f.

As an application of Theorem 10.1.1, we show that some results due to Chatterjee can be expressed in terms of such criterion. From our point of view, this expression seems to ease the original scheme of proof and could possibly lead to various extensions. It also permits to easily recover some known variance bounds in Spin Glass Theory (cf. [5, 6, 11, 12]). Therefore let us present a short introduction to this theory.

Most of the time, in Spin Glasses Theory, it is customary to consider a centered Gaussian field $\big (H_n(\sigma )\big )_{\sigma \in \{-1,1\}^n}$ on the discrete cube {−1, 1}ⁿ (the map σ↦H _n(σ) is called the Hamiltonian of the system) and to focus on $\max _{\sigma \in \{-1,1\}^n}H_n(\sigma )$ (or $\min _{\sigma \in \{-1,1\}^n}H_n(\sigma )$). In general, this quantity is rather complex and presents a lack of regularity. Therefore, one focusses on a smooth approximation of the maximum (or the minimum) called the Free Energy F _n,β. This function is defined as follow

$$\displaystyle \begin{aligned} F_{n,\beta}=\pm\frac{1}{\beta}\log \bigg(\sum_{\sigma\in\{-1,1\}^n}e^{\pm \beta H_n(\sigma)}\bigg) \end{aligned}$$

where β > 0 corresponds to (the inverse of) the temperature and its sign depends on whether you want to study the maximum or the minimum of H _n over the discrete cube.

For instance, for the Random Energy Model (REM in short), we have

$$\displaystyle \begin{aligned} H_n(\sigma)=\sqrt{n}X_\sigma,\quad \sigma\in\{-1,1\}^n \end{aligned}$$

where $(X_\sigma )_{\sigma \in \{-1,1\}^n}$ is a sequence of i.i.d. standard Gaussian random variables.

For the Sherrington and Kirkpatrick’s model (SK model in short), the Hamiltonian is more complex,

$$\displaystyle \begin{aligned} H_n(\sigma)=-\frac{1}{\sqrt{n}}\sum_{i,j=1}^nX_{ij}\sigma_i\sigma_j,\quad \sigma\in\{-1,1\}^n \end{aligned}$$

with (X _ij)_1≤i,j≤n a sequence of i.i.d. standard Gaussian random variables.

As an application of our methodology (cf. Sect. 10.4), we prove the following two Propositions.

Proposition 10.1.1

The following holds for the SK model. Let $0<\beta <\frac {1}{2}$ , then

$$\displaystyle \begin{aligned} \mathrm{Var}(F_{n,\beta})\leq C_\beta,\quad n\geq 1 \end{aligned} $$

(10.2)

where C _β > 0 is a constant depending only on β.

Remark

Talagrand obtained (cf. [11, 12]) such upper bound on the variance, for 0 < β < 1, as a consequence of precise (and much harder to prove than our variance bounds) concentration inequalities for the Free Energy together with second moment method. As far as we know, it is the first time that such bound is obtained through semigroups arguments.

The methodology can also be used for the Random Energy Model (REM in short) (cf. Sect. 10.4 for more details) and provides the following bounds.

Proposition 10.1.2

The following holds in the REM.

High temperature regime: for $0<\beta <\frac {1}{\sqrt {2n}}$ , we have

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_{2^n}}(F_{n,\beta})\leq \frac{n}{2^n} \bigg(\frac{1-n\beta^2}{1-2n\beta^2}\bigg),\quad n\geq 1 \end{aligned}$$

with C > 0 a universal constant.

Remark

• (1) The preceding bound has to be compared with the results exposed in [6, 7] (be careful with the different renormalization, in [6] the free energy is $\frac {F_{n,\beta }}{n}$). In [6], it is shown that

  $$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_{2^n}}(F_{n,\beta})\sim \frac{1}{2^n}\times\frac{e^{n\beta^2}}{\beta^2},\quad \beta<\sqrt{\frac{\log 2}{2}}. \end{aligned}$$

  The dependance (in n and β) is clearly not optimal in this regime but, as presented in Proposition 10.1.1, the scheme of proof of our method is robust enough to treat more complicated models. It seems natural that it can fail to capture precise behaviour such as the one obtained in [6]. Notice also that in [6], the authors obtained various (according to the temperature β) asymptotic convergence results for the (renormalized) Free Energy. Therefore, their results only indicate the correct order of the variance of this functional. However, to our best knowledge, this is the first time that such non-asymptotic bounds on the variance of the Free energy is obtained for the high temperature regime temperature.

• (2) In [6] the low temperature regime was also investigated. Non-asymptotic variance bound, in accordance with the convergence results from Bovier et al., was already obtained in [7] and is presented and commented in Sect. 10.4 (Proposition 10.4.4) for the sake of completeness.

• (3) As we will see latter in this note, it is easier to do the proof (of the preceding result) with the standard Gaussian measure on $\mathbb {R}^n$ and then to perform the following substitutions

  $$\displaystyle \begin{aligned} n\longleftrightarrow 2^n \quad \text{and}\quad \beta\longleftrightarrow \sqrt{n}\beta \end{aligned}$$

  to fit the framework of [6].

This note is organized as follows. In Sect. 10.2, we recall some facts about superconcentration and Gamma calculus. In Sect. 10.3, we will prove our main results. Finally, in Sect. 10.4, we will give some applications in Spin Glass Theory.

10.2 Framework and Tools

In this section, we briefly recall some notions about superconcentration, Gamma calculus and interpolation methods by semigroups. General references about these topics could be, respectively, [1, 7].

10.2.1 Superconcentration

It is well known (cf. [4, 10]), that concentration of measure of phenomenon is useful in various mathematical contexts. Such phenomenon can be obtained through functional inequalities. For instance, the standard Gaussian measure, on $\mathbb {R}^n$, γ _n satisfies a Poincaré’s inequality:

Proposition 10.2.1

For any function $f\,:\,\mathbb {R}^n\to \mathbb {R}$ smooth enough, the following holds

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_n}(f)\leq \int_{\mathbb{R}^n}|\nabla f|{}^2d\gamma_n \end{aligned} $$

(10.3)

where |⋅| stands for the Euclidean norm.

Although this inequality holds for a large class of function, it could lead to sub-optimal bounds. A classical example is the function f(x) =max_i=1,…,nx _i. For such function, Poincaré’s inequality implies that

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_n}(f)\leq 1 \end{aligned}$$

but it is known that $\mathrm {Var}_{\gamma _n}(f)\sim \frac {C}{\log n}$ for some constant C > 0. In Chatterjee’s terminology, in this Gaussian framework, a function f is said to be superconcentrated when Poincaré’s inequality (10.3) is sub-optimal.

As we have said in the introduction, this phenomenon has been studied in various manner: semigroup interpolation [14], Renyi’s representation of order statistics [3], Optimal Transport [15], Ehrard’s inequality [17],…(cf. the Thesis [16] for a recent survey about superconcentration). In this note, we want to show that some differential inequalities between the operator Γ and Γ₂ from Bakry and Émery’s Theory could provide superconcentration.

10.2.2 Semigroups Interpolation and Gamma Calculus

For more details about semigroups interpolation and Γ calculus, we refer to [1, 9]. Although our work can easily be extended to a more general framework, we will focus on a Gaussian setting.

The Ornstein–Uhlenbeck process (X _t)_t≥0 is defined as follow:

$$\displaystyle \begin{aligned}X_t=e^{-t}X+\sqrt{1-e^{-2t}}Y,\quad t\geq 0, \end{aligned}$$

with X and Y  i.i.d. standard Gaussian vectors in $\mathbb {R}^n$. The semigroup (P _t)_t≥0, associated to this process, acts on a class of smooth function $\mathcal {A}$ (due to the integrability of Gaussian densities, one can choose here for $\mathcal {A}$ the class of C ^∞ functions whose derivatives are rapidly decreasing) and admits an explicit representation formula:

$$\displaystyle \begin{aligned}P_tf(x)=\int_{\mathbb{R}^n}f\big(xe^{-t}+\sqrt{1-e^{-2t}}y\big)d\gamma_n(y),\quad x\in\mathbb{R}^n,\,t\geq 0 \end{aligned}$$

Its infinitesimal generator is given by

$$\displaystyle \begin{aligned}L=\Delta-x\cdot\nabla \end{aligned}$$

Furthermore, γ _n is the invariant and reversible measure of (P _t)_t≥0. That is to say, for any function f and g belonging to $\mathcal {A}$,

$$\displaystyle \begin{aligned} \int_{\mathbb{R}^n} P_t fd\gamma_n=\int_{\mathbb{R}^n} fd\gamma_n \quad \text{and}\quad \int_{\mathbb{R}^n} fP_tgd\gamma_n=\int_{\mathbb{R}^n} gP_tfd\gamma_n. \end{aligned}$$

Now, let us recall some properties satisfied by (P _t)_t≥0 which will be useful in the sequel.

Proposition 10.2.2

The Ornstein–Uhlenbeck semigroup (P _t)_t≥0 satisfies the following properties

• P _t(f) is a solution of the heat equation associated to L

  $$\displaystyle \begin{aligned} \mathrm{i.e.}\quad \partial_t (P_tf)=P_t (Lf)=L(P_tf). \end{aligned} $$

  (10.4)

• (P _t)_t≥0 is ergodic, that is to say, for $f\in \mathcal {A}$

  $$\displaystyle \begin{aligned} \lim_{t\to+\infty}P_t(f)= \int_{\mathbb{R}^n}fd\gamma_n=\mathbb{E}_{\gamma_n}[f] \end{aligned} $$

  (10.5)

• (P _t)_t≥0 commutes with the gradient ∇. More precisely, for any function $f\in \mathcal {A}$ ,

  $$\displaystyle \begin{aligned} \nabla P_t(f)=e^{-t}P_t(\nabla f), \quad t\geq 0. \end{aligned} $$

  (10.6)

• (P _t)_t≥0 is a contraction in L ^p(γ _n), for any function f ∈ L ^p(γ _n) and every t ≥ 0,

  $$\displaystyle \begin{aligned} \|P_t(f)\|{}_p\leq \|f\|{}_p. \end{aligned} $$

  (10.7)

As it is exposed in [1], it is possible to give a dynamical representation of the variance of a function f along the semigroup (P _t)_t≥0:

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_n}(f)=2\int_0^\infty \int_{\mathbb{R}^n}|\nabla P_s( f)|{}^2d\gamma_n ds=2\int_0^\infty e^{-2s}\int_{\mathbb{R}^n}|P_s(\nabla f)|{}^2d\gamma_n ds \end{aligned} $$

(10.8)

10.2.3 Gamma Calculus and Poincaré’s Inequality

Let us introduce the fundamental operator Γ₂ and Γ from Bakry and Emery’s Theory. Given an infinitesimal generator L set, for f and g, two smooth functions,

$$\displaystyle \begin{aligned} \Gamma(f,g)=\frac{1}{2}\big[L(fg)-fLg-Lfg\big]\quad \text{and}\quad \Gamma_2(f,g)=\frac{1}{2}\big[L\Gamma(f,g)-\Gamma(f,Lg)-\Gamma(Lf,g)\big] \end{aligned}$$

In the case of the Ornstein–Uhlenbeck’s infinitesimal generator L =  Δ − x ⋅∇, it is easily seen that

$$\displaystyle \begin{aligned} \Gamma(f)=|\nabla f|{}^2\quad \text{and} \quad \Gamma_2(f)=\|\mathrm{Hess} f\|{}_2^2+|\nabla f|{}^2 \end{aligned} $$

(10.9)

where $\|\mathrm {Hess} f\|{ }_2=\big (\sum _{i,j=1}^n\big (\frac {\partial ^2f}{\partial x_i\partial x_j}\big )^2\big )^{1/2}$ is the Hilbert–Schmidt norm of the tensor of the second derivatives of f.

Now, let us briefly recall how a relationship between Γ and Γ₂ can be used to give a elementary proof of Poincaré’s inequality (10.3).

First, notice that the representation formula of the variance (10.8) can be expressed in terms of Γ:

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_n}(f)=2\int_0^\infty \int_{\mathbb{R}^n}\Gamma(P_tf)d\gamma_n ds. \end{aligned} $$

(10.10)

Then, observe that (10.9) implies the celebrated curvature-dimension criterion CD(1, +∞) (cf. [1])

$$\displaystyle \begin{aligned} \Gamma_2\geq \Gamma. \end{aligned} $$

(10.11)

Set $I(t)=\int _{\mathbb {R}^n}\Gamma (P_tf)d\gamma _n$. It is classical that

$$\displaystyle \begin{aligned} I'(t)=-2\int_{\mathbb{R}^n}\Gamma_2(P_tf)d\gamma_n,\quad t\geq 0 \end{aligned}$$

Thus, the inequality (10.11) leads to a differential inequality

$$\displaystyle \begin{aligned} \int_{\mathbb{R}^n}\Gamma_2(P_t f)d\gamma_n\geq \int_{\mathbb{R}^n}\Gamma(P_t f)d\gamma_n \Leftrightarrow 2I+I'\leq0,\quad t\geq 0 \end{aligned} $$

(10.12)

which can be easily integrated between s and t (with 0 ≤ s ≤ t). That is

$$\displaystyle \begin{aligned} I(t)e^{2t}\leq I(s)e^{2s}. \end{aligned}$$

It is now classical to let s → 0 to easily recover Poincaré’s inequality (10.3) for the measure γ _n. As we will see in the next section, we will show that a differential inequality of the form

$$\displaystyle \begin{aligned} I'\geq -2(I+\psi), \end{aligned} $$

(10.13)

for some function ψ, can be used to obtain relevant bound (with respect to superconcentration phenomenon) on the variance of the function f (being fixed) by letting s fixed and t → +∞.

Remark

Let us make few remarks.

• (1) As it is proved in [1], the integrated curvature dimension inequality (10.12) is, in fact, equivalent to the Poincaré’s inequality (10.3).

• (2) As we will see in the next section, the inequality I′≥−2(I + ψ) is equivalent to an inverse, integrated, curvature dimension inequality which seems to be new. However, notice that the major difference between (10.12) and (10.13) is that the first one holds for a large class of function whereas the second is only true for a particular function f (and ψ depends on f).

10.3 Inverse, Integrated, Curvature Inequality

In this section, we will use the methodology exposed in the preceding section to obtain variance bounds for a (fixed) function f satisfying an inverse, integrated, curvature inequality $IC_{\gamma _n}(1,\psi )$.

First, let us state a definition. We want to highlight the fact that this definition will be stated in a Gaussian framework $(\mathbb {R}^n,\Gamma , \gamma _n)$ with Γ associated to the infinitesimal generator L =  Δ − x ⋅∇ and the Ornstein–Uhlenbeck’s semigroup (P _t)_t≥0. The next definition can be extended, mutatis mutandis, to fit the general framework of [1].

Definition 10.3.1

Let $f\,:\,\mathbb {R}^n\to \mathbb {R}$ be a smooth function. We say that f satisfy an inverse, integrated, curvature criterion with function $\psi \,:\,\mathbb {R}_+\to \mathbb {R}$ if

$$\displaystyle \begin{aligned} \int_{\mathbb{R}^n}\Gamma_2(P_t f)d\gamma_n\leq \int_{\mathbb{R}^n}\Gamma(P_t f)d\gamma_n+\psi(t),\quad t\geq 0 \end{aligned} $$

(10.14)

When the previous inequality is satisfied we denote it by $f\in IC_{\gamma _n}(1,\psi )$.

Remark

• (1) Notice, again, that the inequality (10.14) holds, a priori, only for the function f.

• (2) More generally, as it will be needed in the sequel, if μ is a Gaussian measure we will say that f ∈ IC _μ(1, ψ) if Eq. (10.14) is satisfied with μ instead of γ _n and with the operators Γ and Γ₂ associated to the Markov Triple $(\mathbb {R}^n, L,\mu )$.

Now, let us prove our main result Theorem 10.1.1.

Proof (of Theorem 10.1.1)

Assume that $f\in IC_{\gamma _n}(1,\psi )$ (cf. Eq. (10.14)) holds. This is equivalent to the following differential inequality:

$$\displaystyle \begin{aligned} I'\geq -2(I+\psi), \end{aligned} $$

(10.15)

where $I(t)=\int _{\mathbb {R}^n}|\nabla P_t f|{ }^2d\gamma _n$, t ≥ 0. Set I(t) = K(t)e ^−2t, inequality (10.15) becomes

$$\displaystyle \begin{aligned} K'(t)\geq -2e^{2t}\psi(t),\quad t\geq 0 \end{aligned} $$

(10.16)

Now, integrate inequality (10.16) between s and t. That is

$$\displaystyle \begin{aligned}K(t)-K(s)\geq -2\int_s^t e^{2u}\psi(u)du,\quad \text{for all} \quad 0\leq s\leq t. \end{aligned}$$

Then, let t →∞, this yields

$$\displaystyle \begin{aligned}K(s)\leq \big[\lim_{t\to\infty}K(t)\big]+2\int_s^\infty e^{2u}\psi(u)du,\quad s\geq 0, \end{aligned}$$

To conclude, observe that

$$\displaystyle \begin{aligned}K(t)=I(t)e^{2t}\to_{t\to \infty}\bigg|\int_{\mathbb{R}^n}\nabla fd\gamma_n\bigg|{}^2 \end{aligned}$$

by ergodicity of (P _t)_t≥0. Finally, we have, for every t ≥ 0,

$$\displaystyle \begin{aligned} I(t)=\int_{\mathbb{R}_n}\Gamma(P_t f)d\gamma_n\leq e^{-2t} \bigg(\bigg|\int_{\mathbb{R}^n}\nabla fd\gamma_n\bigg|{}^2+2\int_t^\infty e^{2s}\psi(s) ds\bigg). \end{aligned} $$

(10.17)

It suffices to use the dynamical representation of the variance (10.8) with elementary calculus to end the proof. □

Remark

This method of interpolation, between t and + ∞, has also been used in [13] in order to obtain Talagrand’s inequality of higher order.

10.3.1 Another Variance Bound

As we will see in the last section, it is sometimes useful to restrict an IC _μ(1, ψ), for some probability measure μ, up to a time T in order to improve the dependance with respect to some parameter.

In other words, the setting is the following: assume that an IC _μ(1, ψ) holds and that we are able to produce some T > 0 such that the bound of I(T) (given by Eq. (10.17)) is particularly nice (with respect to some parameter). Now, we have to bound the variance in a different manner in order to use the information on I(T). To this task, we will prove the next proposition.

Proposition 10.3.1

Let $f\,:\,\mathbb {R}^n\to \mathbb {R}$ be a function smooth enough. Then, for any T > 0

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_n}(f)\leq \frac{2TI(0)}{1-e^{-2T}}\bigg[\frac{1}{\log a}-\frac{1}{a\log a}\bigg] \end{aligned}$$

with $a=\frac {I(0)}{I(T)}$ and $I(t)=\int _{\mathbb {R}^n}\Gamma (P_tf)d\gamma _n$.

Remark

This proposition will be used to show that the Free Energy is superconcentrated for some Spin Glasses models. Although we stated the preceding Proposition 10.3.1 for the standard Gaussian measure γ _n, it will also hold (up to obvious renormalization) for μ the law of a centered Gaussian vector with covariance matrix M.

To prove the preceding theorem, we will need two further arguments.

First, we present an inequality due to Cordero-Erausquin and Ledoux [8]. The proof of this inequality rests on the fact that the Poincaré’s inequality satisfied by γ _n implies an exponential decay of the variance along the semigroup (P _t)_t≥0.

Lemma 10.3.1 (Cordero-Erausquin–Ledoux)

Let $f\,:\,\mathbb {R}^n\to \mathbb {R}$ be a function smooth enough. Then, for any T > 0, the following holds

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_n}(f)\leq \frac{2}{1-e^{-2T}}\int_0^TI(t) dt \end{aligned} $$

(10.18)

with $I(t)=\int _{\mathbb {R}^n}\Gamma (P_tf)d\gamma _n$.

Proof

For the sake of completeness we give the proof of the preceding Lemma.

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathrm{Var}_{\gamma_n}(f)&\displaystyle =&\displaystyle \mathbb{E}_{\gamma_n}[f^2]-\mathbb{E}_{\gamma_n}[(P_Tf)^2]+\mathbb{E}_{\gamma_n}[(P_Tf)^2]-\mathbb{E}_{\gamma_n}[P_Tf]^2\\ &\displaystyle =&\displaystyle -\int_0^T\frac{d}{ds}\mathbb{E}_{\gamma_n}[(P_sf)^2]ds+\mathrm{Var}_{\gamma_n}(P_Tf)\\ &\displaystyle \leq &\displaystyle 2\int_0^TI(s)ds+e^{-2T}\mathrm{Var}_{\gamma_n}(f). \end{array} \end{aligned} $$

□

Secondly, we will use the fact that the infinitesimal generator (−L) of the Ornstein–Uhlenbeck process (X _t)_t≥0 admits a (discrete) spectral decomposition. Then, denote by dE _λ the spectral resolution of (−L). According to [1], this leads to a different representation of t↦I(t). With $f\,:\,\mathbb {R}^n\to \mathbb {R}$ being fixed, we have:

$$\displaystyle \begin{aligned}I(t)=\int_{\mathbb{R}^n}|\nabla P_tf|{}^2d\gamma_n=\int_0^\infty \lambda e^{-2\lambda t} dE_\lambda(f),\quad t\geq 0 \end{aligned}$$

As it is proven in [2] (cf. Corollary 5.6), t↦I(t) satisfies, with the preceding representation, an Hölder-type inequality. That is to say, for every T > 0,

Lemma 10.3.2 (Baudoin–Wang)

$$\displaystyle \begin{aligned} I(s)\leq I(0)^{1-s/T}I(T)^{s/T},\quad 0\leq s\leq T \end{aligned} $$

(10.19)

Now, we can prove Proposition 10.3.1 with the help of preceding Lemma.

Proof (of Proposition 10.3.1)

First use Lemma 10.3.1 to get

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_n}(f)\leq \frac{2}{1-e^{-2T}}\int_0^TI(t) dt. \end{aligned}$$

Then, use Lemma 10.3.2. This yields

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathrm{Var}_{\gamma_n}(f)&\displaystyle \leq&\displaystyle \frac{2}{1-e^{-2T}}\int_0^TI(0)^{1-t/T}I(T)^{t/T} dt\\ &\displaystyle =&\displaystyle \frac{2I(0)}{1-e^{-2T}}\int_0^Te^{-\frac{t}{T}\log a}dt \end{array} \end{aligned} $$

where $a=\frac { I(0)}{I(T)}\geq 1$ and $I(t)=\int _{\mathbb {R}^n}\Gamma (P_tf)d\gamma _n$. Finally, elementary calculus ends the proof. □

10.4 Application in Spin Glasses’s Theory

In the remaining of this section, we will show how Theorem 10.1.1 can be used to provide relevant bounds on the variance of F _n,β. We will focus on the REM and the SK Model. For the remaining of this note we will denote by f _β, for β > 0, the following function

$$\displaystyle \begin{aligned} f_{\beta}(x)=\frac{1}{\beta}\log \big(\sum_{i=1}^ne^{\beta x_i}\big), \quad x=(x_1,\ldots,x_n)\in\mathbb{R}^n \end{aligned}$$

10.4.1 Random Energy Model

In this section we will show how Theorem 10.1.1 is useful to obtain relevant bound on the variance of the Free Energy F _n,β (with β close to 0) for the REM.

Proposition 10.4.1

For any β > 0, $f_\beta \in IC_{\gamma _n}(1,\psi )$ with

$$\displaystyle \begin{aligned} \psi(t)=2\beta^2e^{-2t} I(t) \end{aligned}$$

where, let us recall it, $I(t)=\int _{\mathbb {R}^n}\Gamma (P_t f_\beta )d\gamma _n$ and Γ is the standard “carré du champ” operator.

We will need the following Lemma to prove the preceding Proposition.

Lemma 10.4.1

Let (u _i)_i=1,…,n be a family of functions, with $u_i\,:\,\mathbb {R}^n\to \mathbb {R}$ for any i = 1, …, n, satisfying the following condition

$$\displaystyle \begin{aligned} \sum_{i=1}^nu_i^2(x)\leq 1\quad \mathit{\text{for all}}\quad x\in \mathbb{R}^n \end{aligned}$$

Then, for any function $v\,:\,\mathbb {R}^n\to \mathbb {R}_+$ and any probability measure μ, we have

$$\displaystyle \begin{aligned}\sum_{i=1}^n\bigg(\int_{\mathbb{R}^n}u_i(x) v(x)d\mu(x)\bigg)^2\leq \bigg(\int_{\mathbb{R}^n}vd\mu\bigg)^2 \end{aligned}$$

Proof

Consider the vector $U=(u_1v,\ldots , u_nv)\in \mathbb {R}^n$ and recall that |⋅| stands for the Euclidean norm. Then, it holds

$$\displaystyle \begin{aligned} \begin{array}{rcl} \bigg[\sum_{i=1}^n\bigg(\int_{\mathbb{R}^n}u_i(x)v(x)d\mu\bigg)^2\bigg]^{1/2}=\bigg|\int_{\mathbb{R}^n}Ud\mu\bigg|\leq \int_{\mathbb{R}^n}|U|d\mu&\displaystyle =&\displaystyle \int_{\mathbb{R}^n}\bigg[\sum_{i=1}^nu_i^2(x)\bigg]^{1/2}v(x)d\mu\\ &\displaystyle \leq &\displaystyle \int_{\mathbb{R}^n}v(x)d\mu \end{array} \end{aligned} $$

where the first upper bound comes from Jensen’s inequality. □

Now we turn to the proof of Proposition 10.4.1.

Proof (of Proposition 10.4.1)

First, observe that the condition $IC_{\gamma _n}(1,\psi )$ is equivalent to

$$\displaystyle \begin{aligned}\int_{\mathbb{R}^n}\Gamma_2\big(P_t(f_\beta)\big)d\gamma_n\leq (1+2\beta^2e^{-2t})\int_{\mathbb{R}^n}\Gamma\Big(P_t (f_\beta)\big)d\gamma_n,\quad t\geq 0. \end{aligned}$$

That is (since $\Gamma _2(f)=\|\mathrm {Hess}f\|{ }_2^2+|\nabla f|{ }^2$ and Γ(f) = |∇f|²)

$$\displaystyle \begin{aligned} \int_{\mathbb{R}_n}\|\mathrm{Hess} P_t(f_\beta)\|{}_2^2d\gamma_n\leq 2\beta^2e^{-2t}\int_{\mathbb{R}^n}|\nabla P_t (f_\beta)|{}^2d\gamma_n,\quad t\geq0. \end{aligned} $$

(10.20)

Now, observe that, pointwise, Eq.(10.20) is equivalent to (thanks to the commutation property between ∇ and (P _t)_t≥0)

$$\displaystyle \begin{aligned}\sum_{i,j=1}^n[P_t(\partial^2_{ij}f_\beta)]^2\leq 2\beta^2\sum_{i=1}^n [P_t(\partial_i f_\beta)]^2,\quad \forall t\geq 0 \end{aligned}$$

Elementary calculus yields, for every i = 1, …, n, and every β > 0,

$$\displaystyle \begin{aligned}\partial_if_\beta=\frac{e^{\beta x_i}}{\sum_{k=1}^n e^{\beta x_k}} \end{aligned}$$

and, for every j = 1, …, n,

$$\displaystyle \begin{aligned}\partial_j\partial_if_\beta=\beta(\partial_if_\beta\delta_{ij}-\partial_if_\beta\partial_j f_\beta). \end{aligned}$$

Thus, for every t ≥ 0,

$$\displaystyle \begin{aligned}\sum_{i,j=1}^n[P_t(\partial^2_{ij}f_\beta)]^2=\beta^2\sum_{i=1}^n\big[P_t(\partial_i f_\beta)\big]^2-2\beta\sum_{i=1}^nP_t(\partial_i f_\beta) P_t\big[(\partial_i f_\beta)^2\big]+\beta^2\sum_{i,j=1}^n\big[P_t(\partial_i f_\beta\partial_j f_\beta)\big]^2. \end{aligned}$$

First ignore the crossed terms (which are always non positive), then apply Lemma 10.4.1 to the third term.

Indeed, let i ∈{1, …, n} be fixed and set u _j = ∂ _jf _β and v = ∂ _if _β. Thus, Lemma 10.4.1 implies

$$\displaystyle \begin{aligned}\sum_{j=1}^n \big[P_t(\partial_if_\beta\partial_jf_\beta)\big]^2\leq P_t^2(\partial_if_\beta). \end{aligned}$$

This inequality finally yields,

$$\displaystyle \begin{aligned}\sum_{i,j=1}^n[P_t(\partial^2_{ij}f_\beta)]^2\leq \beta^2\sum_{i=1}^n\big[P_t(\partial_i f_\beta)\big]^2+\beta^2\sum_{i,j=1}^n\big[P_t(\partial_i f_\beta\partial_j f_\beta)\big]^2 \leq 2\beta^2\sum_{i=1}^n\big[P_t(\partial_i f_\beta)\big]^2.\end{aligned} $$

□

Now, the criterion $IC_{\gamma _n}(1,\psi )$ can be used gives to provide relevant bound on the variance of F _n,β as stated in Proposition 10.1.2.

Proof (of Proposition 10.1.2)

As mentioned earlier, the proof will be done for the standard Gaussian measure on $\mathbb {R}^n$ and then it will be enough to perform a change of variable. As it will be useful in the sequel, observe that (by symmetry) the following holds

$$\displaystyle \begin{aligned} \int_{\mathbb{R}^n}\partial_if_\beta d\gamma_n=\frac{1}{n},\quad \forall i=1,\ldots,n. \end{aligned}$$

Now, let β > 0 and use Theorem 10.1.1 which implies that

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_n}(F_{n,\beta})\leq \frac{1}{n}+4\beta^2\int_0^\infty e^{-2s}(1-e^{-2s})\sum_{i=1}^n\int_{\mathbb{R}^n}P_s^2(\partial_if_\beta)d\gamma_nds \end{aligned} $$

(10.21)

where we used Fubini’s Theorem and the commutation property between ∇ and P _s.

For the first bound, when $\beta \in \big (0,\frac {\sqrt {2}}{2})$, it is possible to rewrite (thanks to the dynamical representation of the variance (10.21)) the integral in the right hand side as

$$\displaystyle \begin{aligned} 2\beta^2 \mathrm{Var}_{\gamma_n}(F_{n,\beta})-4\beta^2\int_0^\infty e^{-4s}\sum_{i=1}^n\int_{\mathbb{R}^n}P_s^2(\partial_if_\beta)d\gamma_nds \end{aligned} $$

(10.22)

Furthermore, by Jensen’s inequality and the invariance of (P _t)_t≥0 with respect to γ _n, we have

$$\displaystyle \begin{aligned} \int_{\mathbb{R}^n}P_s^2(\partial_if_\beta)d\gamma_n\geq \bigg(\int_{\mathbb{R}^n}P_s(\partial_if_\beta)d\gamma_n\bigg)^2=\frac{1}{n^2},\quad \forall i=1,\ldots,n,\quad \forall s>0\end{aligned} $$

Thus, $\mathrm {Var}_{\gamma _n}(F_{n,\beta })\leq \bigg (\frac {1-\beta ^2}{1-2\beta ^2}\bigg )\frac {1}{n}$.

To conclude, as announced, it is enough to substitute n by 2ⁿ and β by $\sqrt {n}\beta$ to get the result. □

Remark

Incidentally, the preceding proof can be used to get a lower bound on the variance of the Free energy. More precisely, it is possible to deduce from (10.21) and (10.22) the following lower bound

$$\displaystyle \begin{aligned} \mathrm{Var}_{\gamma_{2^n}}(F_{n,\beta})\geq \frac{n}{2^n}\frac{(1-n\beta^2)}{(1-2\beta^2n)},\quad \text{for}\quad \beta>\frac{1}{\sqrt{2n}} \end{aligned}$$

10.4.2 SK Model

In this section we show how some work of Chatterjee (from [7]) can be rewritten in term of an inverse, integrated, curvature criterion. Then, it allows us to easily recover a bound, obtained by Talagrand (cf. [11, 12]), on the variance of the Free Energy for the SK model at high temperature.

First, we need to express the Γ and Γ₂ operator when γ _n is replaced by μ the law of a centered Gaussian vector, in $\mathbb {R}^n$, with covariance matrix M.

Let X be a random Gaussian vector with $\mathcal {L}(X)=\mu$ and consider Y  an independent copy of X. It is then possible to define the generalized Ornstein–Uhlenbeck process, which we will still denote by (X _t)_t≥0, as follow

$$\displaystyle \begin{aligned} X_t=e^{-t}X+\sqrt{1-e^{-2t}}Y,\quad t\geq 0\end{aligned} $$

Similarly, we also denote by (P _t)_t≥0 the associated semigroup. Then, it is known (cf. [7, 14, 16]) that, for any smooth function $f\,:\, \mathbb {R}^n \to \mathbb {R}$,

$$\displaystyle \begin{aligned} I(t)=\int_{\mathbb{R}^n}\Gamma(P_t f)d\mu=2\int_{\mathbb{R}^n}e^{-2t}\sum_{i,j}M_{ij}(\partial_if) P_t(\partial_j f)d\mu,\quad t\geq 0\end{aligned} $$

As we will see latter, it will be more convenient to work with

$$\displaystyle \begin{aligned} I_r(t)=2\int_{\mathbb{R}^n}e^{-2t}\sum_{i,j}(M_{ij})^r(\partial_if) P_t(\partial_j f)d\mu,\quad t\geq 0\end{aligned} $$

where r is a positive integer. In the rest of this section, we choose f = f _β.

Proposition 10.4.2 (Chatterjee)

Assume that M _ij ≥ 0 for all (i, j) ∈{1, …, n}². Then, for any t ≥ 0, the following holds

$$\displaystyle \begin{aligned} I^{\prime}_r(t)\geq -2\big[I_r(t)+2\beta^2e^{-2t}J_{r+1}(t)\big] \end{aligned} $$

(10.23)

with J _r(t) = e ^2tI _r(t).

Remark

• (1) In [7], Chatterjee proved that $J^{\prime }_r(t)\geq -4\beta ^2e^{-2t}J_{r+1}(t)$ for any $r\in \mathbb {N}^{\ast }$. The proof is similar the proof of Lemma 10.4.1 with the additional use of Hölder’s inequality.

• (2) In particular, when r = 1, Chatterjee’s proposition amounts of saying that

  $$\displaystyle \begin{aligned} f_\beta\in IC_{\mu}(1,\psi) \end{aligned}$$

  with ψ(t) = 2β ²e ^−2tJ ₂(t). Unfortunately, it remains hard to upper bound this quantity by something relevant.

As observed in the preceding remark, the inverse, integrated, curvature criterion can not be used in the present form. However, it is possible to recycle the arguments of Sect. 10.3. That is, use l times, with $l\in \mathbb {N}$, the fundamental Theorem of analysis (on t↦I _r(t)) together with the inequality (10.23) and let l → +∞. This leads to a useful bound on the function t↦I _r(t) for any $r\in \mathbb {N}^\ast$.

Theorem 10.4.1 (Chatterjee)

Assume that M _ij ≥ 0 for all (i, j) ∈{1, …, n}². Then, for any t ≥ 0, the following holds

$$\displaystyle \begin{aligned} I_r(t)\leq e^{-2t} \sum_{i,j=1}^n(M_{ij})^re^{2\beta^2 e^{-2t}M_{ij}}\nu_i\nu_j,\quad \forall r\geq 1 \end{aligned} $$

(10.24)

where $\nu _i=\int _{\mathbb {R}^n}\partial _if_\beta d\mu$ for all i = 1, …, n.

Remark

When r = 1, the main step of Chatterjee’s proof is equivalent to show that f _β ∈ IC _μ(1, ψ) with $\psi (t)=2\beta ^2e^{-2t}\sum _{i,j=1}M_{ij}e^{2\beta ^2e^{-2t}M_{ij}}\nu _i\nu _j$. The proof of this result can be found in [7, pp. 108–110] .

Unfortunately, the repeated use of the differential inequality (10.23) degrades the upper bound on t↦I _r(t). As we will briefly see in the next subsection, Chatterjee used Eq. (10.24) only for a fixed T > 0 (large enough). We show, in the next Proposition, that this bound (for r = 1) is still relevant to recover some work of Talagrand on the variance of F _n,β, with small β, for the SK model (cf. [11, 12]).

Now, let us prove Proposition 10.1.1.

Proof (of Proposition 10.1.1)

First we show that inequality (10.24) leads to a general upper bound on the variance of F _n,β which might be of independent interest. Then, we choose M to be the covariance structure of the SK model and proved inequality (10.2).

When r = 1, Eq. (10.24) combined with Eq. (10.10) implies that, for any β > 0,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathrm{Var}_\mu(F_{n,\beta})&\displaystyle \leq&\displaystyle 2\int_0^\infty e^{-2t}\sum_{i,j=1}^n M_{ij}e^{2\beta^2e^{-2t}M_{ij}}\nu_i\nu_jdt\\ &\displaystyle \leq&\displaystyle \frac{1}{2\beta^2}\sum_{i,j=1}^ne^{2\beta^2 M_{ij}}\nu_i\nu_j \end{array} \end{aligned} $$

Following Chatterjee (cf. [7]), choose M to be the covariance structure of the SK model. That is,

$$\displaystyle \begin{aligned} M_{\sigma \sigma'}=\bigg(\frac{1}{\sqrt n}\sum_{i=1}^n\sigma_i\sigma_i^{\prime}\bigg)^2,\quad \forall \sigma,\sigma'\in\{-1,1\}^n. \end{aligned}$$

Besides, observe (by symmetry) that, for each σ ∈{−1, 1}ⁿ,

$$\displaystyle \begin{aligned} \nu_{\sigma}=\mathbb{E}_\mu\big[\partial_\sigma F_{n,\beta}\big]=\frac{1}{2^n}. \end{aligned}$$

Thus,

$$\displaystyle \begin{aligned} \mathrm{Var}_\mu(F_{n,\beta})\leq \frac{1}{2\beta^2}\mathbb{E}_{\sigma,\sigma'}\big[e^{2\beta^2\big(\frac{1}{\sqrt{n}}\sigma_i\sigma^{\prime}_i\big)^2}\bigg] \end{aligned}$$

where $\mathbb {E}_{\sigma '\sigma }$ stands for the expectation under the product measure induced by the Rademacher random variables $\sigma _i,\sigma _i^{\prime },\, i=1,\ldots ,n$.

Finally, if $\beta \in \big (0,\frac {1}{2}\big )$ we have $\mathbb {E}_{\sigma ,\sigma '}\big [e^{2\beta ^2\big (\frac {1}{\sqrt {n}}\sum _{i=1}^n\sigma _i\sigma ^{\prime }_i\big )^2}\bigg ]=C(\beta )$. Indeed, observe first that $\sum _{i=1}^n\sigma _i\sigma _i^{\prime }$ has the same distribution as $\sum _{i=1}^n\sigma _i$. Then, it is enough to use Hoeffding’s inequality (cf.[4]), which gives the following deviation inequality

$$\displaystyle \begin{aligned} \mathbb{P}\bigg(\frac{1}{\sqrt{n}}\sum_{i=1}^n\sigma_i>t\bigg)\leq e^{-t^2/2}\quad t\geq 0, \end{aligned}$$

to conclude. □

10.4.3 Improvements of Variance Bounds with Respect to the Parameter β

Let us collect some results of Chatterjee and briefly explain how Proposition 10.3.1 can be used to improve the dependence of the variance bounds with respect to β. However, the dependance in n will be worse.

Chatterjee used, in [7], a Theorem of Bernstein about completely monotone function. As far as we are concerned, the spectral framework exposed in Sect. 10.3 seems to be more natural to work with and provides equivalent results.

The arguments, in order to improve the dependance in β, can be summarize as follow: choose T such that I(T) can be bounded by a relevant quantity and apply Proposition 10.3.1.

Proposition 10.4.3 (Chatterjee)

In the SK model the following holds

$$\displaystyle \begin{aligned} \mathrm{Var}_\mu(F_{n,\beta})\leq \frac{C_1n\log (2+C_2\beta)}{\log n},\quad \forall \beta>0 \end{aligned}$$

with C ₁, C ₂ > 0 two numerical constants.

Remark

Here T > 0 is chosen such that

$$\displaystyle \begin{aligned} \mathbb{E}_{\sigma,\sigma'}\bigg[M_{\sigma\sigma'}e^{2\beta^2e^{-2T}M_{\sigma\sigma'}}\bigg]=C_\beta,\quad \forall \beta>0 \end{aligned}$$

where $M_{\sigma \sigma '}=\big (\frac {1}{\sqrt {n}}\sum _{i=1}^n\sigma _i\sigma _i^{\prime }\big )^2$ and C _β > 0 is a constant that does not depend on n. That is $T=\frac {1}{2}\log \big (\frac {2\beta ^2}{\gamma }\big )$ for some sufficiently small constant γ > 0 (cf. [7]).

Proposition 10.4.4 (Chatterjee)

In the REM, the following holds for $\beta >2\sqrt {\log 2}$ ,

$$\displaystyle \begin{aligned} \mathrm{Var}_\mu(F_{n,\beta})\leq C_\beta \end{aligned}$$

where C _β > 0 is a constant that does not depend on n.

Remark

Here T is chosen as $T=\frac {1}{2}\log (2\beta ^2)$ so that $I(T)\leq \frac {n}{2^n}e^{-2T}e^n$ and the upper bound is relevant in the low temperature regime (cf. [6, 7]). Again, notice the difference of renormalization with Proposition 10.1.2 (one has to replace the number of random variables n by 2ⁿ and the i.i.d. standard Gaussian random variables $(X_i)_{i=1,\ldots ,2^n}$ by $\sqrt {n}X_i$ in the Proposition). In [7], Chatterjee also proved that the upper bound is tight.

In fact, it also possible to use hypercontractive arguments instead of Theorem 10.4.1 to achieve the upper bound of Proposition 10.4.4. Indeed, one can use the inequality (10.21) together with hypercontractive estimates of (P _t)_t≥0 (cf. [7, 8, 15, 16]). More precisely, we have

$$\displaystyle \begin{aligned} \|P_s(\partial_i f_\beta)\|{}_2^2\leq \|\partial_if_\beta\|{}^2_{1+e^{-2s}},\quad \forall i=1,\ldots,n,\quad \forall s>0 \end{aligned}$$

It is then standard, cf. Section 4 in [16] for instance, to prove that

$$\displaystyle \begin{aligned} \int_0^\infty e^{-2s}(1-e^{-2s})\|\partial_if_\beta\|{}_{1+e^{-2s}}^2ds\leq \frac{C\|\partial_i f_\beta\|{}_2^2}{\big[1+\log \frac{\|\partial_i f_\beta\|{}_2}{\|\partial_i f_\beta\|{}_1}\big]^2} \end{aligned}$$

where C > 0 is a numerical constant. Then, it is elementary to conclude. Notice that such estimates are already implicit in the celebrated L ¹∕L ² Talagrand’s inequality (presented in [7, 8] for instance), which one can also be directly used to recover the content of Proposition 10.4.4.