Large Deviation Property of Free Energy in p-Body Sherrington-Kirkpatrick Model

TL;DR

This paper investigates the large deviation properties of free energy in the p-body Sherrington-Kirkpatrick model using the replica method, revealing the behavior of the cumulant generating function and deriving the 1RSB solution.

Contribution

It introduces a novel approach to analyze the cumulant generating function and rate function, and provides a method to derive the 1RSB solution by fixing the RS solution as monotone increasing.

Findings

Cumulant generating function is constant near n=0.

Behavior of phi(n) is derived and shown to break at finite n.

A method to derive the 1RSB solution by fixing the RS solution.

Abstract

Cumulant generating function phi(n) and rate function Sigma(f) of the free energy is evaluated in p-body Sherrington-Kirkpatrick model by using the replica method with the replica number n finite. From a perturbational argument, we show that the cumulant generating function is constant in the vicinity of n = 0. On the other hand, with the help of two analytic properties of phi(n), the behavior of phi(n) is derived again. However this is also shown to be broken at a finite value of n, which gives a characteristic value in the rate function near the thermodynamic value of the free energy. Through the continuation of phi(n) as a function of n, we find out a way to derive the 1RSB solution at least in this model, which is to fix the RS solution to be a monotone increasing function.