Absence of superconcentration for the free energy in the mixed $p$-spin models with external field

Wei-Kuo Chen, Dmitry Panchenko

Published 2015-09-23Version 1

We show that the free energy in the mixed $p$-spin models of spin glasses does not superconcentrate in the presence of external field, which means that its variance is of the order suggested by the Poincar\'e inequality. This complements the result of Chatterjee who showed that the free energy superconcentrates when there is no external field. Our approach is based on a version of Chatterjee's representation for the variance and some consequences of the validity of the Parisi formula for the free energy.