Universality of superconcentration in the Sherrington–Kirkpatrick model

We study the universality of superconcentration for the free energy in the Sherrington–Kirkpatrick model. In [10], Chatterjee showed that when the system consists of (Formula presented.) spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order (Formula presented.), in contrast to the (Formula presented.) bound obtained from the Gaussian–Poincaré 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 (Formula presented.) that arises in the representation of the disorder as (Formula presented.) for (Formula presented.) standard normal. Under an additional regularity assumption on (Formula presented.), we further show that the variance is of order at most (Formula presented.).