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One-sided large deviations for the ground-state energy of spin glasses

Hong-Bin Chen, Alice Guionnet, Justin Ko, Bertrand Lacroix-A-Chez-Toine, Jean-Christophe Mourrat

Abstract

We describe the large deviations above its typical value of the maximal energy of a spin glass with +/-1 spins. Thanks to the relatively explicit description of the rate function we identify, we then show that the latter is asymptotically quadratic near its minimum if and only if an external magnetic field is present. The proof starts from a Parisi-type formula for the fractional moments of the partition function, which we then leverage to obtain the limit of the Laplace transform of the maximum energy. Using convex-duality arguments, we then rewrite this Laplace transform as a supremum over martingales, and thereby deduce the large-deviation principle with explicit rate function.

One-sided large deviations for the ground-state energy of spin glasses

Abstract

We describe the large deviations above its typical value of the maximal energy of a spin glass with +/-1 spins. Thanks to the relatively explicit description of the rate function we identify, we then show that the latter is asymptotically quadratic near its minimum if and only if an external magnetic field is present. The proof starts from a Parisi-type formula for the fractional moments of the partition function, which we then leverage to obtain the limit of the Laplace transform of the maximum energy. Using convex-duality arguments, we then rewrite this Laplace transform as a supremum over martingales, and thereby deduce the large-deviation principle with explicit rate function.
Paper Structure (12 sections, 32 theorems, 215 equations)

This paper contains 12 sections, 32 theorems, 215 equations.

Table of Contents

  1. Introduction and main results
  2. Fractional moments of the partition function
  3. Upper bound on the free energy
  4. Lower bound on the free energy
  5. Laplace transform of the ground-state energy
  6. Un-inverted formula for the Laplace transform
  7. Derivation of the large-deviation principle
  8. Analytic properties of Λ and Λ*
  9. Inverting variations inside Λ*
  10. One-sided large-deviation principle
  11. Proof of Theorem \ref{['t.LDP']}
  12. Necessary and sufficient condition for quadratic behavior

Key Result

Theorem 1.1

We have Moreover, the supremum in e.gs is achieved at a unique $\alpha \in \mathbf{Mart}$.

Theorems & Definitions (69)

  • Theorem 1.1: Un-inverted formula for the ground-state energy
  • Theorem 1.2: Large deviations for the ground-state energy
  • Theorem 1.3: quadratic/non-quadratic behavior near the minimum
  • Theorem 2.1: Fractional moments of $Z_N$
  • Remark 2.2
  • proof : Proof of Theorem \ref{['t.fractional']} from conmin for the SK model with $h_i=0$
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 59 more