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Shattering in Pure Spherical Spin Glasses

Ahmed El Alaoui, Andrea Montanari, Mark Sellke

Abstract

We prove the existence of a shattered phase within the replica-symmetric phase of the pure spherical $p$-spin models for $p$ sufficiently large. In this phase, we construct a decomposition of the sphere into well-separated small clusters, each of which has exponentially small Gibbs mass, yet which together carry all but an exponentially small fraction of the Gibbs mass. We achieve this via quantitative estimates on the derivative of the Franz--Parisi potential, which measures the Gibbs mass profile around a typical sample. Corollaries on dynamics are derived, in particular we show the two-times correlation function of stationary Langevin dynamics must have an exponentially long plateau. We further show that shattering implies disorder chaos for the Gibbs measure in the optimal transport sense; this is known to imply failure of sampling algorithms which are stable under perturbation in the same metric.

Shattering in Pure Spherical Spin Glasses

Abstract

We prove the existence of a shattered phase within the replica-symmetric phase of the pure spherical -spin models for sufficiently large. In this phase, we construct a decomposition of the sphere into well-separated small clusters, each of which has exponentially small Gibbs mass, yet which together carry all but an exponentially small fraction of the Gibbs mass. We achieve this via quantitative estimates on the derivative of the Franz--Parisi potential, which measures the Gibbs mass profile around a typical sample. Corollaries on dynamics are derived, in particular we show the two-times correlation function of stationary Langevin dynamics must have an exponentially long plateau. We further show that shattering implies disorder chaos for the Gibbs measure in the optimal transport sense; this is known to imply failure of sampling algorithms which are stable under perturbation in the same metric.
Paper Structure (17 sections, 28 theorems, 159 equations)

This paper contains 17 sections, 28 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Shattering theorems
  4. Consequences of shattering: I. Langevin dynamics
  5. Consequences of shattering: II. Disorder chaos
  6. Partition function estimates
  7. Planted model and contiguity
  8. Non-monotonicity of Franz--Parisi potential
  9. Construction of the shattered decomposition
  10. Transport disorder chaos
  11. Proof of transport disorder chaos
  12. Lower bound for the onset of disorder chaos
  13. Proof of Theorem \ref{['thm:improved-clustering']}
  14. The static and dynamical temperatures
  15. The static temperature.
  16. ...and 2 more sections

Key Result

Theorem 2.1

For $p$ large enough, there exists a non-empty interval $I_p=(\underline{\beta}(p),\beta_c(p))\subseteq (\beta_d(p),\beta_c(p))$, with $\underline{\beta}(p) = C$ independent of $p$, such that for any $\beta\in I_p=(\underline{\beta}(p),\beta_c(p))$, there exist $\Delta_1,\Delta_2,c>0$ with $\Delta_2

Theorems & Definitions (63)

  • Definition 1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Corollary 2.8
  • ...and 53 more