The method of moments in convolved random matrix models and discrete analogues

Abstract

We study the global and local asymptotics of Macdonald processes, its degenerations, and related models using the method of difference operators. We focus on three applications. First, we consider random plane partitions with interactions, arising from Macdonald processes with periodic weighting. For these models, the Macdonald parameters q; t become interaction parameters for the underlying dimer model. We establish global limit shape and fluctuation theorems in the limit as the mesh size goes to 0 and the interaction parameters tend to 1. Second, we consider a particle system obtained by generalizing the notion of squared singular values of products of truncated orthogonal, unitary, and symplectic matrices to a one-parameter family of deformed models. This procedure is analogous to the extension of classical real, complex, and quaternion matrix ensembles to [beta]-ensembles. A discrete time Markov chain is obtained by considering iterative multiplication of matrix factors and its appropriate generalization. We show global limit shapes and fluctuations when time is of constant order and the number of particles tend to infinity. We also establish local limit theorems at the right edge when time increases with the number of particles. Third, we discuss new developments in the method of difference operators to models beyond Macdonald processes. We apply this technique to obtain moments formulas for eigenvalues of sums of unitarily invariant random matrices and for measures derived from tensor products of representations of the unitary group.