Central Limit Theorems for Cavity and Local Fields of the Sherrington-Kirkpatrick Model

One of the remarkable applications of the cavity method is to prove the Thouless-Anderson-Palmer (TAP) system of equations in the high temperature regime of the Sherrington-Kirkpatrick (SK) model. This naturally leads us to the important study of the limit laws for cavity and local fields. The first quantitative results for both fields based on Stein's method were studied by Chatterjee. Although Stein's method provides us an efficient approach for obtaining the limiting distributions, the nature of this method restricts the derivation of optimal and general results. In this paper, our study based on Gaussian interpolation obtains the CLT for the cavity field. With the help of this result, we conclude the CLT for local fields. In both cases, more refined moment estimates are given.