Phase diagram and large deviations in the free-energy of mean-field spin-glasses

TL;DR

This paper analyzes the large deviations in the free energy of the Sherrington-Kirkpatrick spin-glass model by computing the phase diagram of a key function, confirming scaling laws, and providing exact expressions that match numerical data.

Contribution

It provides a detailed phase diagram of the large deviations function and confirms the $O(n^5)$ scaling at all orders using a hierarchical ansatz.

Findings

Confirmed $O(n^5)$ scaling at all orders

Derived exact expression for the coefficient in terms of $q(x)$

Achieved quantitative agreement with numerical data at zero temperature

Abstract

We consider the probability distribution of large deviations in the spin-glass free energy for the Sherrington-Kirkpatrick mean field model, i.e. the exponentially small probability of finding a system with intensive free energy smaller than the most likely one. This result is obtained by computing $\Phi(n,T)=T \bar{Z^n}/ n$, i.e. the average value of the partition function to the power $n$ as a function of $n$. We study in full details the phase diagram of $\Phi(n,T)$ in the $(n,T)$ plane computing in particular the stability of the replica-symmetric solution. At low temperatures we compute $\Phi(n,T)$ in series of $n$ and $\tau=T_c-T$ at high orders using the standard hierarchical ansatz and confirm earlier findings on the $O(n^5)$ scaling. We prove that the $O(n^5)$ scaling is valid at all orders and obtain an exact expression for the coefficient in term of the function $q(x)$. Resumming the series we obtain the large deviations probability at all temperatures. At zero temperature the analytical prediction displays a remarkable quantitative agreement with the numerical data. A similar computation for the simpler spherical model is also performed and the connection between large and small deviations is discussed.