Rank-One HCIZ Integrals and Multiplicative Convolutions
Abstract
We study how spectral averaging interacts with the logarithm in large-dimensional rank-one Harish-Chandra--Itzykson--Zuber (HCIZ) integrals. For independent bi-unitarily invariant matrices A,B with uniformly bounded deterministic singular values and weakly convergent empirical spectral measures of their Gram matrices, the integral for (BA)*(BA)⊗1N has two limiting descriptions at speed N2 for small real parameters: averaging its logarithm gives free multiplicative convolution, whereas taking the logarithm after averaging gives classical multiplicative convolution. We also prove an exact finite-dimensional equality between this annealed integral and the one for (A*A)⊗(B*B). For every fixed number of factors k≥3, we construct counterexamples showing that the analogous equality can fail even asymptotically. For tensor products with any fixed number k of factors, global Haar vectors at speed Nk and product-state vectors at speed N yield classical and free multiplicative convolution, respectively, for small real parameters under the corresponding hypotheses. We also derive the known multiplicativity of the S-transform.
Related
- Efficient computation of the asymptotics of extensive-rank HCIZ integrals
- Inequalities for rank-two permanents and finite free convolutions
- Finite-rank multiplicative perturbations of rotationally invariant non-Hermitian random matrices
- A Complementarity Property of Rank-One Multiplicative Maps
- A Multiplicative Horn Obstruction to the Unitary Categorification of a Rank-Twelve Modular Datum
BibTeX
@misc{collins2026rankone,
title = {{Rank-One HCIZ Integrals and Multiplicative Convolutions}},
author = {Benoît Collins and Nicolas Delporte and Manasa Nagatsu and Reiko Toriumi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02892},
eprint = {2610.02892},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02892},
}
RIS
TY - UNPB
TI - Rank-One HCIZ Integrals and Multiplicative Convolutions
AU - Benoît Collins
AU - Nicolas Delporte
AU - Manasa Nagatsu
AU - Reiko Toriumi
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02892
UR - https://arxiv.org/abs/2610.02892
AB - We study how spectral averaging interacts with the logarithm in large-dimensional rank-one Harish-Chandra--Itzykson--Zuber (HCIZ) integrals. For independent bi-unitarily invariant matrices A,B with uniformly bounded deterministic singular values and weakly convergent empirical spectral measures of their Gram matrices, the integral for (BA)^*(BA)⊗1_N has two limiting descriptions at speed N^2 for small real parameters: averaging its logarithm gives free multiplicative convolution, whereas taking the logarithm after averaging gives classical multiplicative convolution. We also prove an exact finite-dimensional equality between this annealed integral and the one for (A^*A)⊗(B^*B). For every fixed number of factors k≥3, we construct counterexamples showing that the analogous equality can fail even asymptotically. For tensor products with any fixed number k of factors, global Haar vectors at speed N^k and product-state vectors at speed N yield classical and free multiplicative convolution, respectively, for small real parameters under the corresponding hypotheses. We also derive the known multiplicativity of the S-transform.
ER -