  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.
