On the Discontinuous Breaking of Replica Symmetry and Shattering in Mean-Field Spin Glasses

2025/06/02 by Antonio Auffinger, Auffinger, Antonio, A. El Alaoui +3
Physics and Astronomy · #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2506.02238

Abstract

We show that in mean-field spin glasses, a discontinuous breaking of replica symmetry at the critical inverse temperature β_c implies the existence of an intermediate shattered phase. This confirms a prediction from physics regarding the nature of random first order phase transitions. On the other hand, we give an example of a spherical spin glass which exhibits shattering, yet the transition is continuous at β_c.

Citations

• Dynamics for spherical spin glasses: Gibbs distributed initial conditions
• Near-optimal shattering in the Ising pure p-spin and rarity of solutions returned by stable algorithms
• Disordered Gibbs measures and Gaussian conditioning
• Sampling from Mean-Field Gibbs Measures via Diffusion Processes
• Shattering in Pure Spherical Spin Glasses
• A Spectral Condition for Spectral Gap: Fast Mixing in High-Temperature Ising Models
• Fundamental limits of symmetric low-rank matrix estimation
• Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand bound
• The Aizenman-Sims-Starr scheme and Parisi formula for mixed p-spin spherical models
• Envelope Theorems for Arbitrary Choice Sets

Related

• Shattering Versus Metastability in Spin Glasses
• Introduction to Mean-Field Theory of Spin Glass Models
• Recent rigorous results support the predictions of spontaneously broken replica symmetry for realistic spin glasses
• Shattering in Pure Spherical Spin Glasses
• The State(s) of Replica Symmetry Breaking: Mean Field Theories vs. Short-Ranged Spin Glasses