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

TL;DR

This paper analyzes the fluctuations of the log-partition function in the Sherrington--Kirkpatrick spin glass model near the critical temperature, establishing variance asymptotics and a Gaussian CLT.

Contribution

It provides new variance formulas and a central limit theorem for the model's free energy near criticality, advancing understanding of spin glass fluctuations.

Findings

Variance of free energy scales as -0.5 log(1-β²) near criticality

Variance at critical scaling behaves like (1/6) log N

Centered free energy converges to a Gaussian distribution

Abstract

We consider the Sherrington--Kirkpatrick spin glass model with zero external field and at inverse temperature $\beta>0$. Let $F_N(\beta)$ be the corresponding log-partition function. Under the assumption that $c_N:=N^{1/3}(1-\beta_N^2)$ is bounded away from $0$, we prove that Var$(F_N(\beta_N)) = - \frac{1}{2} \log (1-\beta_N^2) -{\beta_N^2}/{2} + O( c_N^{-3/2}).$ 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 $F_N(\beta_N)$.