Efficient computation of the asymptotics of extensive-rank HCIZ integrals

Antoine Maillard; Jean-Christophe Mourrat

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arXiv · 2609.07771

Efficient computation of the asymptotics of extensive-rank HCIZ integrals

Antoine Maillard·Jean-Christophe Mourrat

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Abstract

We study the high-dimensional asymptotics of Harish-Chandra-Itzykson-Zuber (HCIZ) integrals in the extensive-rank regime. The limit of these integrals is governed by a one-dimensional boundary-value hydrodynamical problem originally derived by Matytsin (1994) and rigorously proved by Guionnet and Zeitouni (2002). Despite its wide-ranging applications, explicit solutions to this problem are known only in a few specific cases. In this work, we introduce an efficient numerical scheme based on a particle discretization and prove its convergence to the continuous boundary-value problem for generic boundary densities. We validate our approach against known analytical solutions and apply it to generic densities, uncovering interesting dynamical phenomena. The high-dimensional limit of HCIZ integrals appears in various contexts, from the large deviations of random matrix spectra to the limiting free energy of disordered systems, high-dimensional statistics, and machine learning. As such, our contribution opens the way towards the numerical exploration of a wide range of high-dimensional models that were previously intractable.

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Antoine Maillard, Jean-Christophe Mourrat. 2026-09-07. Efficient computation of the asymptotics of extensive-rank HCIZ integrals. https://arxiv.org/abs/2609.07771

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