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Fluctuations for the Sherrington--Kirkpatrick spin glass model near the critical temperature

Partha S. Dey, Taegu Kang

Abstract

We consider the Sherrington--Kirkpatrick spin glass model with zero external field and at inverse temperature $β>0$. Let $F_N(β)$ be the corresponding log-partition function. Under the assumption that $c_N:=N^{1/3}(1-β_N^2)$ is bounded away from $0$, we prove that Var$(F_N(β_N)) = - \frac{1}{2} \log (1-β_N^2) -{β_N^2}/{2} + O( c_N^{-3/2}).$ As a consequence, we obtain Var$(F_N(1-c N^{-1/3})) = \frac16\log N + O(1)$ for any fixed constant $c\in(0,\infty)$. We also prove a Gaussian central limit theorem for the centered and scaled $F_N(β_N)$.

Fluctuations for the Sherrington--Kirkpatrick spin glass model near the critical temperature

Abstract

We consider the Sherrington--Kirkpatrick spin glass model with zero external field and at inverse temperature . Let be the corresponding log-partition function. Under the assumption that is bounded away from , we prove that Var As a consequence, we obtain Var for any fixed constant . We also prove a Gaussian central limit theorem for the centered and scaled .
Paper Structure (11 sections, 15 theorems, 95 equations)

This paper contains 11 sections, 15 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Coupled Hamiltonians and variance representation
  4. Stein's method for normal approximation
  5. Cavity method
  6. Proofs
  7. Proof of Theorem \ref{['thm:main']}
  8. Ingredients for the proof of Proposition \ref{['prop:R^2 main estimate']}
  9. Proof of Proposition \ref{['prop:R^2 main estimate']} part I
  10. Proof of Proposition \ref{['prop:R^2 main estimate']} part II
  11. Acknowledgements.

Key Result

Theorem 1.1

Assume that the inverse temperature sequence $(\beta_N)_{N \geqslant 1}$ satisfies $\lim_{N\to\infty}\beta_N=1$ and Define the function for $\beta<1$. Then, the variance satisfies Moreover, we have the following distributional convergence for the free energy Here, $L$ is a finite constant depending on the sequence $(\beta_N)_{N \geqslant 1}$, $g\sim\mathop{\mathrm{N}}\nolimits(0,1)$ and $\math

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1
  • Remark 2
  • Lemma 1.3: Cha09
  • Lemma 1.4
  • Lemma 1.5: Tal1
  • Proposition 1.6
  • Corollary 2.1
  • proof
  • ...and 16 more