Central limit theorems for cavity and local fields of the Sherrington-Kirkpatrick model

One of the remarkable applications of the cavity method in the mean field spin glasses is to prove the validity of the Thouless-Anderson-Palmer (TAP) system of equations in the Sherrington-Kirkpatrick (SK) model in the high temperature regime. This naturally leads us to the study of the limit laws for cavity and local fields. The first quantitative results for both fields were obtained by Chatterjee [1] using Stein's method. In this paper, we approach these problems using the Gaussian interpolation technique and establish central limit theorems for both fields by giving moment estimates of all orders.