On the Tail of the Overlap Probability Distribution in the Sherrington--Kirkpatrick Model

TL;DR

This paper analyzes the large deviation behavior of the overlap probability density in the Sherrington--Kirkpatrick model across different phases, combining analytical methods and numerical simulations to understand its tail behavior.

Contribution

It provides a comprehensive analytical study of the overlap distribution's tail in the SK model, including corrections near the critical temperature and exact results in the paramagnetic phase.

Findings

The overlap distribution tail behaves as approximately -A((|q|-q_{EA})^3) in the spin glass phase.

First correction to the tail behavior near the critical temperature is computed.

Numerical simulations confirm the analytical predictions on moderate lattice sizes.

Abstract

We investigate the large deviation behavior of the overlap probability density in the Sherrington--Kirkpatrick model from several analytical perspectives. First we analyze the spin glass phase using the coupled replica scheme. Here generically $\frac1N \log P_N(q)$ $\approx$ $- {\cal A}$ $((|q|-q_{EA})^3$, and we compute the first correction to the expansion of $\A$ in powers of $T_c-T$. We study also the $q=1$ case, where $P(q)$ is know exactly. Finally we study the paramagnetic phase, where exact results valid for all $q$'s are obtained. The overall agreement between the various points of view is very satisfactory. Data from large scale numerical simulations show that the predicted behavior can be detected already on moderate lattice sizes.