Asymptotics of rectangular spherical integrals

In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain the asymptotics of the so-called rectangular spherical integrals as m, n go to infinity while m/n converges to a positive constant. We also improve and give a new proof of the asymptotics for the Harish-Chandra-Itzykson-Zuber integral derived in [37,38]. (c) 2023 Elsevier Inc. All rights reserved.