A tail bound for multi-matrix models with Lipschitz interaction
Abstract
We give non-asymptotic operator-norm tail bounds for random multi-matrix models with quadratic plus globally Lipschitz interactions. More precisely, let μf be the probability measure on (𝕄n)sam with density proportional to exp(−n2(lVert x rVert22/2+f(x))). If f is L-Lipschitz with respect to lVert · rVert1, we construct a coupling of X∼μf and an m-tuple Z of independent GUE matrices such that lVert X−Z rVert∞≤ L almost surely, where lVert · rVert∞ is the maximum of the operator norms of the matrix tuple. This is an operator-norm analog of the L∞-Wasserstein bound of Khudiakova, Maas, and Pedrotti (2025). We give a new proof based on the variational formula of Boué and Dupuis (1998), the associated Hamilton--Jacobi--Bellman equation, and the duality between entropy and pressure.
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BibTeX
@misc{diez2026tail,
title = {{A tail bound for multi-matrix models with Lipschitz interaction}},
author = {Charles-Philippe Diez and David Jekel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05660},
eprint = {2610.05660},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05660},
}
RIS
TY - UNPB
TI - A tail bound for multi-matrix models with Lipschitz interaction
AU - Charles-Philippe Diez
AU - David Jekel
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05660
UR - https://arxiv.org/abs/2610.05660
AB - We give non-asymptotic operator-norm tail bounds for random multi-matrix models with quadratic plus globally Lipschitz interactions. More precisely, let μ_f be the probability measure on (𝕄_n)_sa^m with density proportional to exp(−n^2(lVert x rVert_2^2/2+f(x))). If f is L-Lipschitz with respect to lVert · rVert_1, we construct a coupling of X∼μ_f and an m-tuple Z of independent GUE matrices such that lVert X−Z rVert_∞≤ L almost surely, where lVert · rVert_∞ is the maximum of the operator norms of the matrix tuple. This is an operator-norm analog of the L^∞-Wasserstein bound of Khudiakova, Maas, and Pedrotti (2025). We give a new proof based on the variational formula of Boué and Dupuis (1998), the associated Hamilton--Jacobi--Bellman equation, and the duality between entropy and pressure.
ER -