Free energy fluctuations in SK and related spin glass models: A literature survey

TL;DR

This survey reviews the progress made over 50 years in understanding free energy fluctuations in SK and related spin glass models, highlighting key results across temperature regimes and external field conditions.

Contribution

It compiles and discusses the extensive literature on free energy fluctuations in SK and spherical SK models, emphasizing recent advances and remaining challenges.

Findings

Progress in characterizing free energy fluctuations

Results across different temperature regimes

Insights into phase transitions

Abstract

Over the past 50 years, spin glass models have generated a broad range of literature in mathematics, physics, and computer science. There has been much progress in characterizing and proving the limiting free energy of various models, stemming from the original formulas of Parisi. Comparatively less is known about the more detailed topic of free energy fluctuations. This paper concerns a family of models in which there has been considerable progress on fluctuations, namely the Sherrington-Kirkpatrick (SK) and spherical Sherrington-Kirkpatrick (SSK) models, along with their multi-species analogs. We present a survey of the literature on free energy fluctuations in these 2-spin models, discussing results from different temperature regimes, with and without an external field, including results on phase transitions.