Spin glasses and the Parisi formula

2025/10/01 by Mourrat, Jean-Christophe
#Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR)

paper · doi:10.48550/arxiv.2510.01054

Abstract

Spin glasses are models of statistical mechanics in which a large number of simple elements interact with one another in a disordered fashion. One of the fundamental results of the theory is the Parisi formula, which identifies the limit of the free energy of a large class of such models. Yet many interesting models remain out of reach of the classical theory, and direct generalizations of the Parisi formula yield invalid predictions. I will report here on some partial progress towards the resolution of this problem, which also brings a new perspective on classical results.

Citations

• The convex structure of the Parisi formula for multi-species spin glasses
• Balanced multi-species spin glasses
• On free energy of non-convex multi-species spin glasses
• An informal introduction to the Parisi formula
• Potential Hessian Ascent: The Sherrington-Kirkpatrick Model
• Injective norm of real and complex random tensors I: From spin glasses to geometric entanglement
• Statistical mechanics of mean-field disordered systems: a Hamilton-Jacobi approach
• Un-inverting the Parisi formula
• Breakdown of a concavity property of mutual information for non-Gaussian channels
• Nobel Lecture: Multiple equilibria
• Algorithmic Threshold for Multi-Species Spherical Spin Glasses
• Infinite-dimensional Hamilton-Jacobi equations for statistical inference on sparse graphs
• Mutual information for the sparse stochastic block model
• Hamilton-Jacobi equations from mean-field spin glasses
• TAP approach for multi-species spherical spin glasses I: general theory
• TAP approach for multi-species spherical spin glasses II: the free energy of the pure models
• Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses
• Statistical inference of finite-rank tensors
• Free energy upper bound for mean-field vector spin glasses
• Optimization of Mean-field Spin Glasses
• The Overlap Gap Property and Approximate Message Passing Algorithms for\n p-spin models
• Optimization of the Sherrington-Kirkpatrick Hamiltonian
• Following the ground-states of full-RSB spherical spin glasses
• Free energy of bipartite spherical Sherrington--Kirkpatrick model
• Community Detection and Stochastic Block Models
• Statistical physics of inference: thresholds and algorithms
• Proof of the satisfiability conjecture for large k
• Satisfiability threshold for random regular NAE-SAT
• Maximum independent sets on random regular graphs
• Multi-species mean-field spin-glasses. Rigorous results
• On the equivalence of Hopfield Networks and Boltzmann Machines
• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• Learning Patterns and Pattern Sequences by Self-Organizing Nets of Threshold Elements
• Statistical mechanics of neural networks near saturation

Related

• An informal introduction to the Parisi formula
• On Parisi measures of Potts spin glasses with correction
• The convex structure of the Parisi formula for multi-species spin glasses
• A Constructive Proof of the Spherical Parisi Formula
• Parisi's formula is a Hamilton-Jacobi equation in Wasserstein space