Title: Asymptotic expansion for noncommutative smooth functions of multimatrix models
Abstract: A multimatrix model is an ensemble of d-tuples Y^N = (Y_1^N,…,Y_d^N) of NxN Hermitian random matrices with joint density proportional to exp(-N Tr(V^N(X))) for some (convex) NxN Hermitian matrix–valued function V^N of d-tuples of NxN Hermitian matrices. In this case, V^N is called the potential of the model. This talk is based on joint work with D. Jekel and F. Parraud in which we study the large-N behavior of multimatrix models with potential satisfying certain bounds on its second derivatives. In particular, we give an asymptotic expansion in powers of 1/N^2 of the trace of “noncommutative smooth functions” of Y^N and, as a consequence, establish strong convergence for such models. Our proof is based on the construction of transport maps between the laws of different multimatrix models and then the asymptotic expansion of those transport maps.
Reza Gheissari
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