Efficient computation of the asymptotics of extensive-rank HCIZ integrals
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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BibTeX
@misc{maillard2026efficient,
title = {{Efficient computation of the asymptotics of extensive-rank HCIZ integrals}},
author = {Antoine Maillard and Jean-Christophe Mourrat},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07771},
eprint = {2609.07771},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05771867v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Efficient computation of the asymptotics of extensive-rank HCIZ integrals
AU - Antoine Maillard
AU - Jean-Christophe Mourrat
PY - 2026
DA - 2026/09/07
PB - HAL
DO - 10.48550/arxiv.2609.07771
UR - https://hal.science/hal-05771867v1
AB - 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.
ER -