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

Elizabeth Collins-Woodfin, Han Gia Le

TL;DR

This literature survey addresses free-energy fluctuations in 2-spin mean-field spin glasses, focusing on SK and SSK and their multi-species variants. It synthesizes methods ranging from Gaussian interpolation, cluster expansions, and Parisi theory to contour-integral and random-matrix techniques, to characterize fluctuations across high, low, and critical temperatures and under external fields or Curie-Weiss terms. Key findings include Gaussian fluctuations in the high-temperature SK regime, GOE Tracy-Widom fluctuations for SSK at low temperature, and various transitional regimes under external fields, CW interactions, and multi-species heterogeneity, with rigorous bounds and, in several cases, explicit limiting distributions. The results illuminate the deep connections between spin-glass fluctuations and random-matrix theory, RS/RSB phases, and phase-transition phenomena, providing a comprehensive map that guides future work in more complex or non-mean-field settings.

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.

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

TL;DR

This literature survey addresses free-energy fluctuations in 2-spin mean-field spin glasses, focusing on SK and SSK and their multi-species variants. It synthesizes methods ranging from Gaussian interpolation, cluster expansions, and Parisi theory to contour-integral and random-matrix techniques, to characterize fluctuations across high, low, and critical temperatures and under external fields or Curie-Weiss terms. Key findings include Gaussian fluctuations in the high-temperature SK regime, GOE Tracy-Widom fluctuations for SSK at low temperature, and various transitional regimes under external fields, CW interactions, and multi-species heterogeneity, with rigorous bounds and, in several cases, explicit limiting distributions. The results illuminate the deep connections between spin-glass fluctuations and random-matrix theory, RS/RSB phases, and phase-transition phenomena, providing a comprehensive map that guides future work in more complex or non-mean-field settings.

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.

Paper Structure

This paper contains 32 sections, 18 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Scope of the survey
  3. Model set-up
  4. Sherrington--Kirkpatrick (SK) model
  5. Spherical Sherrington--Kirkpatrick (SSK) model
  6. Ground state energy
  7. SK (or SSK) with an external field
  8. SK (or SSK) with ferromagnetic/Curie-Weiss interaction
  9. Multi-species models
  10. Conventions
  11. Some foundational results
  12. Brief remarks on literature beyond this survey
  13. Organization of the paper
  14. Free energy fluctuations in the SK model
  15. Results for SK model without external field
  16. ...and 17 more sections

Key Result

Theorem 2.1

For all $\beta<1$, we have the convergence in distribution as $N \to \infty$ where $X \sim \mathcal{N}(-\frac{1}{2} \sigma^2, \sigma^2)$ and $\sigma^2=-\frac{1}{2}\log(1-\beta^2)-\frac{\beta^2}{2}$.

Figures (1)

  • Figure 1: Phase diagram for SSK with ferromagnetic Curie-Weiss interaction, adapted from BLW18.

Theorems & Definitions (25)

  • Theorem 2.1: SK at high temperature AizenmanLebowitzRuelle
  • Remark 2.2
  • Theorem 2.3: Upper bound under finite temperature, Chatterjee09
  • Conjecture 2.4
  • Theorem 2.5: SK model at the critical temperature, ChenLam19
  • Theorem 2.6: SK model with external field, ChenDeyPanchenko
  • Theorem 2.7: SK model at high temp, with weak external field DeyWu23
  • Remark 2.8
  • Remark 2.9
  • Theorem 2.10: SK model with CW interaction Banerjee2020
  • ...and 15 more