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The Two Point Function of the SK Model without External Field at High Temperature

Christian Brennecke, Adrien Schertzer, Changji Xu, Horng-Tzer Yau

Abstract

We show that the two point correlation matrix $ \textbf{M}= (\langle σ_i σ_j\rangle)_{1\leq i,j\leq N} $ of the Sherrington-Kirkpatrick model with zero external field satisfies \[ \lim_{N\to\infty} \| \textbf{M} - ( 1+β^2 - β\textbf{G})^{-1} \|_{\text{op}} =0 \] in probability, in the full high temperature regime $β< 1$. Here, $\textbf{G}$ denotes the GOE interaction matrix of the model.

The Two Point Function of the SK Model without External Field at High Temperature

Abstract

We show that the two point correlation matrix of the Sherrington-Kirkpatrick model with zero external field satisfies in probability, in the full high temperature regime . Here, denotes the GOE interaction matrix of the model.
Paper Structure (4 sections, 13 theorems, 196 equations, 3 figures)

This paper contains 4 sections, 13 theorems, 196 equations, 3 figures.

Table of Contents

  1. Introduction and Main Result
  2. Setup and Reduction Step
  3. Proof of Theorem \ref{['thm:main']}
  4. Proof of the Identity \ref{['eq:heur2']}

Key Result

Theorem 1.1

Assume that $\beta < 1$ and denote by $\textbf{P} = (p_{ij})_{1\leq i,j\leq N}\in \mathbb{R}^{N\times N}$ the matrix whose diagonal entries are equal to one and whose off-diagonal entries are given by for $i\neq j$, where $g_e := g_{i_1i_2}$ for edges $e = \{i_1,i_2\}$. Then it holds true that in the sense of probability. In particular, we have that

Figures (3)

  • Figure 1:
  • Figure 2: Example of two distinct pairs $(\gamma,\tau), (\gamma', \tau')\in \Gamma_{\text{loop} }\times \Gamma_{\text{sc}} \in S_{\eta}^{ij}$ for some $\eta \in \Gamma_{\text{loop} }$ such that $\psi_2(\gamma\circ\tau) = \psi_2(\gamma'\circ\tau')$. Note that (b) is obtained from (a) by simply switching the roles of $\gamma$ and $\tau$.
  • Figure 3: Example graphs related to Step 3: $\gamma\in\Gamma_{\text{loop} }$ is colored in blue, $\tau\in \Gamma_{\text{sc}}$ is depicted in black, $\{i,j\}$ is colored in green and we denote $\psi_1 = \psi_1(\gamma\circ\tau), \psi_2 = \psi_2(\gamma\circ\tau)$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 15 more