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 $μ_f$ be the probability measure on $(\mathbb{M}_n)_{\operatorname{sa}}^m$ with density proportional to $\exp(-n^2(\lVert x \rVert_2^2/2+f(x)))$. If $f$ is $L$-Lipschitz with respect to $\lVert \cdot \rVert_1$, we construct a coupling of $X\simμ_f$ and an $m$-tuple $Z$ of independent GUE matrices such that $\lVert X-Z \rVert_\infty\leq L$ almost surely, where $\lVert \cdot \rVert_\infty$ is the maximum of the operator norms of the matrix tuple. This is an operator-norm analog of the $L^\infty$-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.