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Simultaneous replica-symmetry breaking for vector spin glasses

Hong-Bin Chen, Jean-Christophe Mourrat

TL;DR

The paper analyzes large-$N$ mean-field vector spin glasses with possibly non-convex interactions, showing that, up to small perturbations, the limit Gibbs measure is governed by a critical point of an explicit Hamilton–Jacobi functional and that replica-symmetry breaking is simultaneous across all spin types.A core tool is the continuous Poisson–Dirichlet cascade, implemented via a cascade of Gaussian processes with covariance given by the overlap function, together with an enriched free energy and a Parisi PDE encoding the ultrametric structure.The authors develop a robust decomposition of matrix-valued overlap paths into Lipschitz and quantile components, define canonical and joint decompositions, and establish a cascade-based Parisi PDE representation with sharp regularity and stability results.Using PDE characteristics, endpoint analyses, and a mechanism that transfers increments in the Parisi measure across species through coupling in the interaction function, the work proves simultaneous RSB for vector and multi-species spin glasses and discusses explicit coupling scenarios.

Abstract

We consider mean-field vector spin glasses with possibly non-convex interactions. Up to a small perturbation of the parameters defining the model, the asymptotic behavior of the Gibbs measure is described in terms of a critical point of an explicit functional. In this paper, we study some properties of these critical points. Under modest assumptions ensuring that different types of spins interact, we show that the replica-symmetry-breaking structures of the different types of spins are in one-to-one correspondence with one another. For instance, if some type of spins displays one level of replica-symmetry breaking, then so do all the other types of spins. This extends the recent results of [Electronic Journal of Probability, 27:1-75, 2022] and [Comm. Math. Phys., 394(3):1101-1152, 2022] that were obtained in the case of multi-species spherical spin glasses with convex interactions.

Simultaneous replica-symmetry breaking for vector spin glasses

TL;DR

The paper analyzes large-$N$ mean-field vector spin glasses with possibly non-convex interactions, showing that, up to small perturbations, the limit Gibbs measure is governed by a critical point of an explicit Hamilton–Jacobi functional and that replica-symmetry breaking is simultaneous across all spin types.A core tool is the continuous Poisson–Dirichlet cascade, implemented via a cascade of Gaussian processes with covariance given by the overlap function, together with an enriched free energy and a Parisi PDE encoding the ultrametric structure.The authors develop a robust decomposition of matrix-valued overlap paths into Lipschitz and quantile components, define canonical and joint decompositions, and establish a cascade-based Parisi PDE representation with sharp regularity and stability results.Using PDE characteristics, endpoint analyses, and a mechanism that transfers increments in the Parisi measure across species through coupling in the interaction function, the work proves simultaneous RSB for vector and multi-species spin glasses and discusses explicit coupling scenarios.

Abstract

We consider mean-field vector spin glasses with possibly non-convex interactions. Up to a small perturbation of the parameters defining the model, the asymptotic behavior of the Gibbs measure is described in terms of a critical point of an explicit functional. In this paper, we study some properties of these critical points. Under modest assumptions ensuring that different types of spins interact, we show that the replica-symmetry-breaking structures of the different types of spins are in one-to-one correspondence with one another. For instance, if some type of spins displays one level of replica-symmetry breaking, then so do all the other types of spins. This extends the recent results of [Electronic Journal of Probability, 27:1-75, 2022] and [Comm. Math. Phys., 394(3):1101-1152, 2022] that were obtained in the case of multi-species spherical spin glasses with convex interactions.

Paper Structure

This paper contains 17 sections, 35 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Continuous cascades
  3. Definition and properties of the continuous cascade
  4. Enriched free energy
  5. Decomposition of matrix-valued paths
  6. Quantile functions
  7. Decompositions
  8. Decomposition of one path
  9. Canonical decompositions
  10. Joint decomposition of multiple paths
  11. The Parisi PDE
  12. Proof of the main result
  13. Computation along characteristics
  14. Left endpoints of paths
  15. Detection of increments
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose that the assumptions e.ass.full.support and e.ass.coupling hold. Let $t > 0$, $q \in \mathcal{Q}_1$, and let $(q',p) \in \mathcal{Q}_{\infty}^2$ be a critical point of $\mathcal{J}_{t,q}$. For every $s < s' \in [0,1]$, if $p(s') - p(s) \neq 0$, then $p(s') - p(s) \in S^D_{++}$.

Theorems & Definitions (74)

  • Theorem 1.1: Simultaneous RSB
  • Lemma 2.1: Invariance of cascades
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • ...and 64 more