Critical Free-Energy Variance in the Sherrington-Kirkpatrick Model

Abstract

We study the variance of the free energy in the Gaussian Sherrington-Kirkpatrick model at zero external field. For every system size and inverse temperature $\beta >0$, we prove $2N^2\mathrm{Var}{F}_N^{\mathrm{off}}(\beta )\le \log \chi +({\beta }^2-1)(\chi -1),\chi =N\mathbb{E}⟨R_{1,2}^2{⟩}_{\beta }$, where $R_{1,2}$ is the replica overlap and $F_N^{\mathrm{off}}=N^{-1}\log Z_N^{\mathrm{off}}$ omits diagonal disorder. The proof uses an exact identity for the edges incident to one spin, Schur complements, and Gaussian disorder interpolation. At criticality the bound reduces to $2N^2\mathrm{Var}{F}_N^{\mathrm{off}}(1)\le \log \chi$. An explicit nonnegative integral measures its gap. A second decomposition relates this gap to a mixed cubic overlap functional. Existing moment and cavity estimates imply that its required integral is bounded. Critical susceptibility asymptotics give the sharp logarithmic upper coefficient and show that the gap converges exactly when the variance has a constant term. We extend an entropy bound throughout bounded critical temperature windows and establish overlap moment stability on a smaller interval.

arXiv comment32 pages, v2: 37 pages. Extended finite-size bounds and strengthened remainder estimates. Updated references