Universality of Superconcentration in the Sherrington-Kirkpatrick Model

Abstract: We study the universality of superconcentration for the free energy in the Sherrington-Kirkpatrick (SK) model. In arXiv:0907.3381, Chatterjee showed that when the system consists of $N$ spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order $N/\log{N}$, in contrast to the $O(N)$ bound obtained from the Gaussian-Poincar\'e inequality. In this paper, we show that superconcentration indeed holds for any choice of centered disorders with finite third moment, where the upper bound is expressed in terms of an auxiliary nondecreasing function $f$ that arises in the representation of the disorder as $f(g)$ for $g$ standard normal. Under an additional regularity assumption on $f$, we further show that the variance is of order at most $N/\log{N}$.