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The Threshold Energy of Low Temperature Langevin Dynamics for Pure Spherical Spin Glasses

Mark Sellke

Abstract

We study the Langevin dynamics for spherical $p$-spin models, focusing on the short time regime described by the Cugliandolo-Kurchan equations. Confirming a prediction of [Cugliandolo-Kurchan, Phys. Rev. Lett. 1993], we show the asymptotic energy achieved is exactly $E_{\infty}(p)=2\sqrt{\frac{p-1}{p}}$ in the low temperature limit. The upper bound uses hardness results for Lipschitz optimization algorithms and applies for all temperatures. For the lower bound, we prove the dynamics reaches and stays above the lowest energy of any approximate local maximum. In fact the latter behavior holds for any Hamiltonian obeying natural smoothness estimates, even with disorder-dependent initialization and on exponential time-scales.

The Threshold Energy of Low Temperature Langevin Dynamics for Pure Spherical Spin Glasses

Abstract

We study the Langevin dynamics for spherical -spin models, focusing on the short time regime described by the Cugliandolo-Kurchan equations. Confirming a prediction of [Cugliandolo-Kurchan, Phys. Rev. Lett. 1993], we show the asymptotic energy achieved is exactly in the low temperature limit. The upper bound uses hardness results for Lipschitz optimization algorithms and applies for all temperatures. For the lower bound, we prove the dynamics reaches and stays above the lowest energy of any approximate local maximum. In fact the latter behavior holds for any Hamiltonian obeying natural smoothness estimates, even with disorder-dependent initialization and on exponential time-scales.
Paper Structure (22 sections, 27 theorems, 136 equations)

This paper contains 22 sections, 27 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. The Threshold Energy $E_{\infty}$
  3. Our Results
  4. Statement of the Lower Bound
  5. On the Proof of Theorem \ref{['thm:mainLB-general']}
  6. Statement of the Upper Bound
  7. On Mixed $p$-Spin Models
  8. Notations and Technical Preliminaries
  9. Estimates for $1$-Dimensional Diffusions
  10. Lower Bound: Proof of Theorem \ref{['thm:mainLB']}
  11. A Priori Estimates and Energy Gain Away From Approximate Critical Points
  12. Stereographic Projection
  13. Quadratic Potential Approximation
  14. Energy Gain at Quadratic Saddle Points
  15. Upper Bound: Proof of Theorem \ref{['thm:mainUB']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

For any $p\geq 2$ and $\eta>0$, for some $T_0=T_0(p,\eta)$ and all sufficiently large $\beta\geq \beta_0(p,\eta)$, for $N$ sufficiently large and any (possibly $H_N$-dependent) ${\boldsymbol{x}}_0\in{\mathcal{S}}_N$:

Theorems & Definitions (59)

  • Theorem 1.1
  • Definition 1
  • Definition 2
  • Theorem 1.2
  • Lemma 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • ...and 49 more