The Kac Version of the Sherrington–Kirkpatrick Model at High Temperatures

We study the Kac version of the Sherrington–Kirkpatrick (SK) model of a spin glass, i.e., a spin glass with long- but finite-range interaction on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}^d $$ \end{document} and Gaussian mean zero couplings. We prove that for all β < 1, the free energy of this model converges to that of the SK model as the range of the interaction tends to infinity. Moreover, we prove that for all temperatures for which the infinite-volume Gibbs state is unique, the free energy scaled by the square root of the volume converges to a Gaussian with variance cγ, β, where γ−1 is the range of the interaction. Moreover, at least for almost all values of β, this variance tends to zero as γ goes to zero, the value in the SK model. We interpret our finding as a weak indication that at least at high temperatures, the SK model can be seen as a reasonable asymptotic model for lattice spin glasses.