Keating Jonathan P, Wong Mo Dick
Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG UK.
Department of Mathematical Sciences, Durham University, Stockton Road, Durham, DH1 3LE UK.
Commun Math Phys. 2022;394(3):1247-1301. doi: 10.1007/s00220-022-04429-3. Epub 2022 Jun 29.
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.
我们研究了低于临界维度的次临界高斯乘性混沌（GMCs）的“临界时刻”。特别地，我们建立了一个关于主导阶渐近性的完全显式公式，它与GMCs的大偏差结果密切相关，并展示了类似的普遍性特征。我们推测，我们的结果正确地描述了随机矩阵矩的类似矩的行为，或者更一般地说，在整个介观尺度上渐近高斯且对数相关的结构的行为。在循环酉系综的设定下，对于整数情形这一推测得到了验证，从而将Claeys等人以及Fahs的结果扩展并强化到了高阶矩。
