Concentration of Multi-overlaps for Random Dilute Ferromagnetic Spin Models

Abstract

We consider mean field ferromagnetic spin models on dilute random graphs and prove that, with suitable one-body infinitesimal perturbations added to the Hamiltonian, the multi-overlaps concentrate for all temperatures, both with respect to the thermal Gibbs average and the quenched randomness. Results of this nature have been known only for the lowest order overlaps, at high temperature or on the Nishimori line. Here we treat all multi-overlaps by a non-trivial application of Griffiths–Kelly–Sherman correlation inequalities. Our results apply in particular to the pure and mixed p-spin ferromagnets on random dilute Erdoes–Rényi hypergraphs. On physical grounds one expects that multi-overlap concentration is an important ingredient for the validity of the cavity (or replica-symmetric) formula for the pressure of mean field models. However rigorously establishing this formula for the p-spin ferromagnet on a random dilute hypergraph remains an open problem.

Appendices

Some Technicalities

1.1 Proof of the Approximation Inequality (7)

Note that

$$\begin{aligned} \big | p_{n}(h_0, h_1, \alpha ) - p_{n}(0,0,0) \big |&= \big | p_{n}(h_0, h_1, \alpha ) - p_{n}(0,h_1,0) \big | \\&\le \big | p_{n}(h_0, h_1, \alpha ) - p_{n}(0, h_1, \alpha ) \big |\\&\quad + \big | p_{n}(0, h_1, \alpha ) - p_{n}(0, h_1, 0) \big |\,. \end{aligned}$$

We have $| \frac{d p_{n}(h_0, h_1, \alpha )}{d{h_0}}| = |{\mathbb {E}}\langle Q_1\rangle | \le 1$ and from (23) we also have $| \frac{d p_n(0, h_1, \alpha )}{d\alpha }| \le h_1 n^{-(1-\theta )}$. Thus by the mean value theorem we obtain (7), i.e. $|p_n(h_0,h_1,\alpha ) - p_n(0,0,0)| \le h_0 + \alpha h_1n^{-(1-\theta )}$.

1.2 A Property of the Poisson Distribution

Any function $g: \mathbb {N} \rightarrow \mathbb {R}$ of a random variable $X\sim \mathrm{Poi}(\nu )$ with Poisson distribution and mean $\nu$, and such that both $\mathbb {E}\,g(X)$ and $\mathbb {E}\,g(X+1)$ exist, satisfies

$$\begin{aligned} \frac{d \, \mathbb {E}\,g(X)}{d \nu }&= \sum _{k=0}^{\infty } \frac{d}{d\nu }\bigg \{\frac{\nu ^{k} e^{-\nu }}{k!}\bigg \} g(k) = \sum _{k=1}^{\infty } \frac{\nu ^{k-1} e^{-\nu }}{(k-1)!} g(k) - \sum _{k=0}^{\infty } \frac{\nu ^k e^{-\nu }}{k!} g(k) \nonumber \\&= \sum _{k=0}^{\infty } \frac{\nu ^{k} e^{-\nu }}{k!} g(k + 1) - \sum _{k=0}^{\infty } \frac{\nu ^k e^{-\nu }}{k!} g(k) = \mathbb {E}\,g(X+1) - \mathbb {E}\, g(X)\,. \end{aligned}$$

(45)

1.3 Multivariate Harris Inequality

For completeness we provide here a simple proof of the multivariate version of the Harris inequality. We refer to [39] for more information.

Lemma 12

(Multivariate version of the Harris inequality) Let $g,\tilde{g}:\mathbb {R}^n\mapsto \mathbb {R}$ be two functions of the random vector $\mathbf{x }= (x_1, \ldots , x_n)$ where all components are independent random variables. If for all $i \in \{1, \ldots , n\}$g and $\tilde{g}$ are both monotone w.r.t. $x_i$ with same monotonicity, i.e. $\partial _{x_i}g(\mathbf{x })\,\partial _{x_i}\tilde{g}(\mathbf{x })\ge 0 \ \forall \ i$, then $\mathbb {E}[g(\mathbf{x })\, \tilde{g}(\mathbf{x })] - \mathbb {E}\,g(\mathbf{x }) \, \mathbb {E}\,\tilde{g}(\mathbf{x }) \ge 0$.

Proof

Let $\mathbf{x }_i^j \equiv (x_i, x_{i+1}, \ldots , x_{j})$. The monotonicity w.r.t. $x_1$ implies

$$\begin{aligned} \mathbb {E}_{x_1} \mathbb {E}_{x'_1} \big [ \big ( g(x_1, \mathbf{x }_2^n) - g(x'_1, \mathbf{x }_2^n) \big ) \big ( \tilde{g}(x_1, \mathbf{x }_2^n) - \tilde{g}(x'_1, \mathbf{x }_2^n) \big ) \big ] \ge 0 \end{aligned}$$

which by expanding the product can be simplified to

$$\begin{aligned} \mathbb {E}_{x_1}[g(\mathbf{x }) \,\tilde{g}(\mathbf{x })]-\mathbb {E}_{x_1}g(\mathbf{x })\, \mathbb {E}_{x_1}\tilde{g}(\mathbf{x }) \ge 0\,. \end{aligned}$$

The proof then proceeds by induction. Suppose

$$\begin{aligned} \mathbb {E}_{\mathbf{x }_1^{i-1}}[g(\mathbf{x })\, \tilde{g}(\mathbf{x })]-\mathbb {E}_{\mathbf{x }_1^{i-1}}g(\mathbf{x })\, \mathbb {E}_{\mathbf{x }_1^{i-1}}\tilde{g}(\mathbf{x }) \ge 0\,. \end{aligned}$$

(46)

Again, the monotonicity w.r.t. $x_i$ implies

$$\begin{aligned}&\mathbb {E}_{x_i} \mathbb {E}_{x'_i} \big [ \big ( \mathbb {E}_{\mathbf{x }_1^{i-1}} g(\mathbf{x }_1^{i-1}, x_i, \mathbf{x }_{i+1}^n) - \mathbb {E}_{\mathbf{x }_1^{i-1}} g(\mathbf{x }_1^{i-1}, x'_i, \mathbf{x }_{i+1}^n) \big ) \\&\quad \cdot \big ( \mathbb {E}_{\mathbf{x }_1^{i-1}} \tilde{g}(\mathbf{x }_1^{i-1}, x_i, \mathbf{x }_{i+1}^n) - \mathbb {E}_{\mathbf{x }_1^{i-1}} \tilde{g}(\mathbf{x }_1^{i-1}, x'_i, \mathbf{x }_{i+1}^n) \big ) \big ] \ge 0 \end{aligned}$$

which can be simplified to

$$\begin{aligned} \mathbb {E}_{x_i} \big [ \mathbb {E}_{\mathbf{x }_1^{i-1}} g(\mathbf{x }) \, \mathbb {E}_{\mathbf{x }_1^{i-1}} \tilde{g}(\mathbf{x }) \big ]-\mathbb {E}_{\mathbf{x }_1^{i}}g(\mathbf{x })\, \mathbb {E}_{\mathbf{x }_1^{i}}\tilde{g}(\mathbf{x }) \ge 0\,. \end{aligned}$$

(47)

The induction is ended by noting that with the hypothesis (46) the identity (47) can further be relaxed to

$$\begin{aligned} \mathbb {E}_{\mathbf{x }_1^{i}}[g(\mathbf{x })\, \tilde{g}(\mathbf{x })]-\mathbb {E}_{\mathbf{x }_1^{i}}g(\mathbf{x })\, \mathbb {E}_{\mathbf{x }_1^{i}}\tilde{g}(\mathbf{x }) \ge 0\,. \end{aligned}$$

This ends the induction argument and the proof. $\square$

On the Concentration and Existence of the Pressure

We consider Hamiltonian (1) with independent random couplings $J_X$, $X\subset \{1,\ldots , n\}$ and prove the following generic result used in (2). We then discuss a simple argument and condition that guarantees the existence of the thermodynamic limit using the first GKS inequality. We verify that these results apply to Example 1.

Proposition 13

(Concentration of the pressure) Let $J_X$, $X\subset \{1,\ldots , n\}$ be independent random variables such that $\sum _{X\subset \{1,\ldots , n\}} \mathrm {Var}(J_X) \le C_P n$ for some numerical constant $C_P>0$. Then we have $\mathbb {E}[(P_n - p_n)^2] \le C_P/n$.

Proof

The proof is a simple application of the Efron-Stein inequality. Set $\varvec{J}\equiv (J_X, X\subset \{1,\ldots ,n\})$. Let $\varvec{J}^{(X)}$ be a vector such that $\varvec{J}^{(X)}$ differs from $\varvec{J}$ only at the Xth component which becomes $J_X^\prime$ drawn independently from the same distribution as the one of $J_X$ (note that the random variables $J_X$ for different X do not necessarily have the same distribution). Efron Stein’s inequality tells us that

$$\begin{aligned} \mathbb {E} \big [ ( P_n - \mathbb {E}\,P_n )^2 \big ] \le \frac{1}{2} \sum _{X\subset \{1,\ldots ,n\}} \mathbb {E}_{\varvec{J}\setminus J_X} \mathbb {E}_{J_X}\mathbb {E}_{J'_X} \big [ ( P_n(\varvec{J}) - P_n(\varvec{J}^{(X)}) )^2 \big ] \,. \end{aligned}$$

(48)

An elementary interpolation gives

$$\begin{aligned} \big | P_n(\varvec{J}) - P_n(\varvec{J}^{(X)})\big |&= \frac{1}{n} \Big | \int _0^1 ds \frac{d}{ds} \ln \sum _{\varvec{\sigma } \in \{\pm 1 \}^n} \exp \big \{- \mathcal {H}_0(\varvec{\sigma },\varvec{J}^{(X)}) + s (J_X -J'_X) \sigma _X \big \} \Big | \\&= \frac{1}{n} \Big | \int _0^1 ds\, (J_X - J'_X) \langle \sigma _X \rangle _s \Big | \le \frac{1}{n} | J_X - J'_X | \,. \end{aligned}$$

Replacing in (48) (and recalling $p_n\equiv \mathbb {E}\, P_n$) gives

$$\begin{aligned} \mathbb {E}\big [(P_n - p_n)^2\big ] \le \frac{1}{2n^2} \sum _{X\subset \{1,\ldots ,n\}} \mathbb {E}_{J_X} \mathbb {E}_{J'_X} \big [ ( J_X - J'_X )^2 \big ] = \frac{1}{n^2}\sum _{X\subset \{1,\ldots ,n\}}\mathrm {Var}(J_X)\,. \end{aligned}$$

With the hypothesis on $\mathrm {Var}(J_X)$ the proof is complete. $\square$

An easy and more or less standard superadditivity argument proves that the thermodynamic limit exists for the ferromagnetic model (1). We give the argument for completeness. For simplicity we consider that there exists a maximal size $x_\mathrm{max}$ independent of n such that $\vert X\vert \le x_\mathrm{max}$. We suppose furthermore that all $J_X$ are independent with a distribution that depends only on the cardinalities $\vert X\vert$ (in other words given a cardinality they are i.i.d.) and also

$$\begin{aligned} \frac{1}{n}\sum _{X\subset \{1, \ldots , n\}}\mathbb {E}\,J_X = \frac{1}{n}\sum _{\vert X\vert =1}^{x_\mathrm{max}}\left( {\begin{array}{c}n\\ \vert X\vert \end{array}}\right) m(\vert X\vert ) \le C \end{aligned}$$

(49)

where $m({\vert X\vert }) \equiv \mathbb {E}\,J_X$ and C a positive constant independent of n.

Proposition 14

(Existence of the thermodynamic limit of the pressure) Let $J_X, X \subset \{1,\ldots ,n\}$ be independent random variables with a probability distribution supported on $\mathbb {R}_{\ge 0}$ depending only on $\vert X\vert$. Moreover assume $J_X=0$ for $\vert X\vert > x_\mathrm{max}$ independent of n. Let (49) be satisfied. Then $\lim _{n \rightarrow +\infty } p_n$ exists and is finite.

Proof

Fix non-zero integers $n_1$, $n_2$ both greater than $x_\mathrm{max}$ and $n \equiv n_1 + n_2$. Consider a set of realizations $S\equiv \{J_X, X\subset \{1,\ldots ,n\}\}$. This set can be split in three disjoint sets $S=S_1\cup S_2\cup S_{12}$ with $S_1 \equiv \{J_X, X\subset \{1,\ldots , n_1\}\}$, $S_2 \equiv \{J_X, X\subset \{n_1+1,\ldots , n\}\}$ and $S_{12} \equiv \{J_X, X\cap \{1,\ldots , n_1\} \ne \emptyset , X\cap \{n_1+1,\ldots , n\} \ne \emptyset \}$. Let $\ln {\mathcal {Z}}(S)/n$ the pressure corresponding to the Hamiltonian with couplings in S and $\ln {\mathcal {Z}}(S_1)/n_1$ and $\ln {\mathcal {Z}}(S_2)/n_2$ the pressures corresponding to the Hamiltonians with couplings from $S_1$ and $S_2$ only. One can show, using the first GKS inequality, that⁴

$$\begin{aligned} \ln {\mathcal {Z}}(S) \ge \ln {\mathcal {Z}}(S_1) + \ln {\mathcal {Z}}(S_2)\,. \end{aligned}$$

Then averaging over all coupling constants in S, using that they are independent with distributions depending only on the cardinality $\vert X\vert$ and that all cardinalities are contained in S, $S_1$ and $S_2$, we obtain

$$\begin{aligned} \mathbb {E}_S\ln {\mathcal {Z}}(S) \ge \mathbb {E}_{S_1}\ln {\mathcal {Z}}(S_1) + \mathbb {E}_{S_2}\ln {\mathcal {Z}}(S_2) \end{aligned}$$

which is equivalent to $np_n \ge n_1 p_{n_1} + n_2 p_{n_2}$ (for $n_1$, $n_2$ greater than $x_\mathrm{max}$). This means that the function $n\mapsto np_n$ is a superadditive sequence and therefore by Fekete’s lemma the limit $\lim _{n\rightarrow +\infty } p_n$ equals $\sup _n p_n$. To show that $\sup _n p_n$ is finite note that

$$\begin{aligned} p_n \le - \frac{1}{n} \,\mathbb {E}\,\min _{\varvec{\sigma }}\mathcal {H}(\varvec{\sigma }) + \ln 2 = \frac{1}{n} \sum _{X\subset \{1,\ldots ,n\}} \mathbb {E}\,J_X + \ln 2 \le C+ \ln 2 \end{aligned}$$

using $J_X\ge 0$ and condition (49). This ends the proof. $\square$

Consider now Example 1 for n large and p fixed. We have $J_X = 0$ for all subsets with cardinalities $\vert X\vert$ different from 1 and p. For $\vert X\vert =1$ the coupling constants $J_X=H$ are deterministic so obviously $\mathrm {Var}(J_X) =0$. For $\vert X\vert =p$ the couplings $J_X$ are independent Bernoulli variables taking value J with probability $\gamma n \left( {\begin{array}{c}n\\ p\end{array}}\right) ^{-1}$ and 0 with complementary probability, so $\mathrm {Var}(J_X) = J^2 \gamma n \left( {\begin{array}{c}n\\ p\end{array}}\right) ^{-1} \big ( 1 - \gamma n \left( {\begin{array}{c}n\\ p\end{array}}\right) ^{-1} \big )$. Thus

$$\begin{aligned} \sum _{X\subset \{1,\ldots ,n\}}\mathrm {Var}(J_X) = \left( {\begin{array}{c}n\\ p\end{array}}\right) J^2 \gamma n \left( {\begin{array}{c}n\\ p\end{array}}\right) ^{-1} \Big ( 1 - \gamma n \left( {\begin{array}{c}n\\ p\end{array}}\right) ^{-1} \Big ) < J^2\gamma n\,. \end{aligned}$$

Therefore Proposition 13 applies. Similarly the condition for the existence of the thermodynamic limit of the pressure is also met because the left hand side of (49) equals

$$\begin{aligned} \frac{1}{n}\sum _{X\subset \{1,\ldots ,n\}}\mathbb {E}\,J_X = H + J\gamma \,. \end{aligned}$$

The mixed p-spin models can be treated similarly.