Signatures of interference in statistical mechanics of local random circuits

A sufficiently long local random circuit generates a quantum chaotic evolution operator. Quantum chaos is ultimately described by ensembles of random matrixes. How close the evolution operator is to such a universal behavior can be diagnosed by comparing moments of operators up to order k to their values in a random matrix ensemble. Ensemble average moments are described by statistical mechanics models, in which the matrix elements are predicted by Weingarten calculus. In this talk we address the question of how the characteristics of the underlying speckle pattern manifest in the statistical models. We report theoretical and numerical analysis of such moments and identify features reflecting the interference phenomena.