A note on Lata\la's argument in SK model

Abstract

In this note, we consider the Sherrington--Kirkpatrick model with deterministic external field. Let $q=q(\beta,h)$ denote the solution of the replica-symmetric self-consistency equation \[ q=\mathbb E\tanh^2\!\left(h+\beta\sqrt q\,Z\right), \qquad Z\sim N(0,1), \] where $\beta$ and $h$ are inverse temperature and external field, respectively. By refining Lata\la's argument, previously limited to \(\beta<\frac{1}{2}\), and using the Kearns--Saul inequality, we prove overlap concentration and convergence of the free energy to the replica symmetric formula with error \(O(N^{-1})\) whenever \[ \beta^2\frac{q}{{\rm arctanh}q}<1. \] Note that for any $\beta<1$ and $h\in \mathbb R$, the condition above is satisfied. Moreover, for every nonzero $h$, this region contains a nonempty interval with $\beta>1$.