Overlap Fluctuations from Random Overlap Structures

TL;DR

This paper explores overlap fluctuations in the Sherrington-Kirkpatrick spin glass model using Random Overlap Structures, revealing that Aizenman-Contucci polynomials naturally emerge and impose restrictions consistent with Parisi theory.

Contribution

It introduces an iterative method to demonstrate the emergence of AC polynomials in Boltzmann ROSts for almost all temperatures, extending to general ROSts including Parisi's structure.

Findings

AC polynomials arise in Boltzmann ROSts at almost all temperatures

Overlap fluctuations are constrained by AC polynomials in agreement with Parisi theory

Results extend to any ROSt, including the Parisi structure

Abstract

We investigate overlap fluctuations of the Sherrington-Kirkpatrick mean field spin glass model in the framework of the Random Over- lap Structure (ROSt). The concept of ROSt has been introduced recently by Aizenman and coworkers, who developed a variational approach to the Sherrington-Kirkpatrick model. We propose here an iterative procedure to show that, in the so-called Boltzmann ROSt, Aizenman-Contucci (AC) polynomials naturally arise for almost all values of the inverse temperature (not in average over some interval only). The same results can be obtained in any ROSt, including therefore the Parisi structure. The AC polynomials impose restric- tions on the overlap fluctuations in agreement with Parisi theory.