The high temperature region of the Viana-Bray diluted spin glass model

TL;DR

This paper analyzes the high temperature phase of the Viana-Bray diluted spin glass model, establishing annealing, fluctuation limits, and divergence of variance at the replica-symmetric boundary, with implications for the SK model.

Contribution

It provides a comprehensive analysis of the replica symmetric phase of the Viana-Bray model, including free energy control and fluctuation behavior, extending understanding of diluted spin glasses.

Findings

Proves annealing for the infinite volume free energy.

Establishes a central limit theorem for multi-overlap fluctuations.

Shows free energy fluctuations converge to a non-Gaussian variable with diverging variance at the boundary.

Abstract

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.