Sherrington-Kirkpatrick Mean Field Spin Glass Research Articles

On ℓp-Gaussian–Grothendieck Problem

Abstract For $p\geq 1$ and $(g_{ij})_{1\leq i,j\leq n}$ being a matrix of i.i.d. standard Gaussian entries, we study the $n$-limit of the $\ell _p$-Gaussian–Grothendieck problem defined as $$\begin{align*} & \max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}. \end{align*}$$The case $p=2$ corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when $p=\infty $, the maximum value is essentially the ground state energy of the Sherrington–Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $1\leq p<2$ and $2<p<\infty .$ For the former, we compute the limit of the $\ell _p$-Gaussian–Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order of $n^{-1}$.

On the Almeida-Thouless transition line in the Sherrington-Kirkpatrick model with centered Gaussian external field

We study the phase transition of the free energy in the Sherrington-Kirkpatrick mean-field spin glass model with centered Gaussian external field. We show that the corresponding Almeida-Thouless line is the correct transition curve that distinguishes between the replica symmetric and replica symmetry breaking solutions in the Parisi formula.

Order of fluctuations of the free energy in the SK model at critical temperature

We present an elementary approach to the order of fluctuations for the free energy in the Sherrington-Kirkpatrick mean field spin glass model at and near the critical temperature. It is proved that at the critical temperature the variance of the free energy is of $O((\log N)^2).$ In addition, we show that if one approaches the critical temperature from the low temperature regime at the rate $O(N^{-\alpha})$ for some $\alpha>0,$ then the variance is of $O((\log N)^2+N^{1-\alpha}).$

Overlap fluctuations from the Boltzmann random overlap structure

We investigate overlap fluctuations of the Sherrington-Kirkpatrick mean field spin glass model in the framework of the Random Overlap Structure (ROSt). The concept of ROSt has been introduced recently by Aizenman and co-workers, who developed a variational approach to the Sherrington-Kirkpatrick model. Here we propose an iterative procedure to show that, in the so-called Boltzmann ROSt, Aizenman-Contucci polynomials naturally arise for almost all values of the inverse temperature (not in average over some interval only). These polynomials impose restrictions on the overlap fluctuations in agreement with Parisi theory.

The High Temperature Region of the Viana–Bray Diluted Spin Glass Model

In this paper, we study the high temperature or low connectivity phase of the Viana-Bray model. This is a diluted version of the well known Sherrington-Kirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy, and a central limit theorem for the suitably rescaled fluctuations of the multi-overlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a non-Gaussian random variable, whose variance diverges at the boundary of the replica-symmetric region. The connection with the fully connected Sherrington-Kirkpatrick model is discussed.

Quadratic replica coupling in the Sherrington–Kirkpatrick mean field spin glass model

We develop a very simple method to study the high temperature, or equivalently high external field, behavior of the Sherrington–Kirkpatrick mean field spin glass model. The basic idea is to couple two different replicas with a quadratic term, trying to push out the two replica overlap from its replica symmetric value. In the case of zero external field, our results reproduce the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, we can prove the validity of the Sherrington–Kirkpatrick replica symmetric solution up to a line, which falls short of the Almeida–Thouless line, associated to the onset of the spontaneous replica symmetry breaking, in the Parisi ansatz. The main difference with the method, recently developed by Michel Talagrand, is that we employ a quadratic coupling, and not a linear one. The resulting flow equations, with respect to the parameters of the model, turn out to be very simple, and the parameter region, where the method works, can be easily found in explicit terms. As a straightforward application of cavity methods, we show also how to determine free energy and overlap fluctuations, in the region where replica symmetry has been shown to hold. It is a major open problem to give a rigorous mathematical treatment of the transition to replica symmetry breaking, necessarily present in the model.

ABOUT THE OVERLAP DISTRIBUTION IN MEAN FIELD SPIN GLASS MODELS

We continue our presentation of mathematically rigorous results about the Sherrington-Kirkpatrick mean field spin glass model. Here we establish some properties of the distribution of overlaps between real replicas. They are in full agreement with the Parisi accepted picture of spontaneous replica symmetry breaking. As a byproduct, we show that the self-averaging of the Edwards-Anderson fluctuating order parameter, with respect to the external quenched noise, implies that the overlap distribution is given by the Sherrington-Kirkpatrick replica symmetric Ansatz. This extends previous results of Pastur and Scherbina. Finally, we show how to generalize our results to realistic short range spin glass models.