On the Critical-Subcritical Moments of Moments of Random Characteristic Polynomials: A GMC Perspective.

We study the 'critical moments' of subcritical Gaussian multiplicative chaos (GMCs) in dimensions . In particular, we establish a fully explicit formula for the leading order asymptotics, which is closely related to large deviation results for GMCs and demonstrates a similar universality feature. We conjecture that our result correctly describes the behaviour of analogous moments of moments of random matrices, or more generally structures which are asymptotically Gaussian and log-correlated in the entire mesoscopic scale. This is verified for an integer case in the setting of circular unitary ensemble, extending and strengthening the results of Claeys et al. and Fahs to higher-order moments.