Entropy and dimension spectrum of the mean-field spin glass model

A self-contained new constructive solution of the mean-field Sherrington-Kirkpatric spin-glass model is obtained from the study of the behaviour of the entropy of the Gibbs measure at low temperatures.


Introduction
In two recent papers [1,2] we studied the freezing property for a class of logarithmically correlated Gaussian fields.In particular, we have shown that the freezing phenomenon can be characterized by the vanishing of the entropy of the associated Gibbs measures at low temperatures.This is demonstrated for several models models widely studied in statistical physics (random polymers on the trees [3,4], Multiplicative Chaos [5], Random Energy Model [6], spin-glass models, etc).In the case of Sherrington-Kirpatrick we have proved that the entropy of the Gibbs measure vanishes beyond a "freezing" inverse temperature β f and obtained a sharp bound for the low temperature limiting free energy.In the following, using the method introduced in these works and without making use of the replica method, we present a new solution of the low temperature free energy and discuss the behaviour of the entropy and fractal dimension of the Gibbs measure.

Definitions and main result
We briefly recall the definition of the model.WIth each site from a finite set of n sites, we associate the one-spin space Σ := {1, −1}.The configuration space is given by the product space Σ n = Σ n = {−1, 1} n , of card Σ n = 2 n .For each σ ∈ Σ n , the finite volume Hamiltonian of the model is defined by the following real-valued function on Σ n where the couplings J = (J ij ) 1≤i<j≤n are independent centered Gaussian variables of variance one.
At the inverse temperature β = 1 T > 0 (we take units such that the Boltzmann constant k = 1), the disorder-dependent partition function Z n (β, J), is given by the sum of the Boltzmann factors The entropy of µ n,β (σ) is defined by S(µ n,β ) = − σ µ n,β (σ) log µ n,β (σ).The functions denote the quenched average of the specific free energy and the annealed specific free energy respectively.In particular, f ∞ is self-averaging (i.e.non random.)The thermodynamic limits f ∞ (β) and f ∞ (β) exist and are given by The phase transition of the model occurs at the inverse temperature β c defined by We define moreover the freezing temperature β f of the model by The main result of the present work is summarized in the following Theorem 1.For all β ≥ β 1 ≡ 1, assuming the differentiability of the free energy with respect to the inverse temperature, the limit f ∞ (β), is given almost surely by where s(µ β 1 |µ β ) is the relative entropy of the measure µ β 1 with respect tho the measure µ β .
One can remark that the previous solution is much more legible that the replica symmetry breaking one [9].In the next section we show that it is obtained by making use of the variational principle and convexity arguments.Moreover, the estimation of the relative entropy allows us to describe the behaviour of the graph of the limiting free energy at low temperatures.

Proof of the main result
We first recall that, for all β > 0, the quenched limit f ∞ (β) exists and is a convex function of β [7].Let β 1 ≡ 1.From the high temperature results [8], we have, almost surely, that Thus, for β = β 1 , the quenched limit f ∞ (β 1 ) equals to the annealed one f ∞ (β 1 ) = β 2 1 /4 + log 2, where the term β 2  1 /4 comes from the mean value of the Boltzmann factor, i.e. the typical behaviour and the mean behaviour coincide at this temperature.
Moreover, from the definition of the specific entropy, we remark that the limiting free energy is given by where s(µ β ) = lim n→∞ 1 n S(µ n,β ).We introduce now the relative entropy density s(µ β 1 |µ β ) of the probability mesure µ β 1 w.r.t. the probability measure µ β , defined by We recall that the relative entropy is a non-negative function, vanishing in case the two measures are equal, and quantifies the extent by which the measure µ β 1 "differs" from the measure µ β .
Using this definition we get for all β ≥ β 1 : (where we have used the fact , we obtain that the limiting free energy given a.s.by The previous formula is much simpler than the solution obtained from the complicated one given by the replica symmetry method [9] and it is intuitively appealing.Indeed, one can recognize that the first two terms correspond to the convexity bound and the third term (the relative entropy) represents the deviation of the Gibbs measure µ β 1 from the measure µ β .
By recalling now the geometric fact that each point of the graph of a convex function lies above its sub-differential, we obtain, for all β ≥ 1, a sharp lower bound for the graph of the limit f ∞ (β).This is given by the dashed red line in figure 1.
Figure 1.Behaviour of the lower bound of the limiting free energy f ∞ (β) for all positive temperatures.Up to β 1 , the lower bound is indeed the exact limit; beyond β 1 , the quenched limit is strictly smaller than the annealed one (whose graph is depicted by the dotted black curve.)The dashed red line represents the lower bound.Beyond the value β * the specific entropy is zero and the free energy is linear.For β 1 ≤ β ≤ β * , the limiting free energy is a convex function whose lower bound is depicted by the dashed red curve passing necessarily from B (see text.) We can now draw some conclusions on the Hausdorff dimension of the support of the sets of configurations supporting the Gibbs measure.Since the specific entropy of the Gibbs measure vanishes for all β ≥ β * , one can conclude that the Hausdorff dimension of the support of the Gibbs measure is strictly positive for all β ≤ β * and vanishes beyond the β * .Indeed, the socalled Parisi replica symmetry breaking region corresponds indeed to the temperature interval [1, β * ] where freezing is happening on smaller and smaller configuration scales.For β ≥ β * , the system is frozen and the measure is essentially dominated by the local extreme values of the random interactions J ij .

Acknowledgments
Partial support from the Labex MME-DII is gratefully acknowledged.