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 ${\mu }_f$ be the probability measure on $({\mathbb{M}}_n{)}_{\mathrm{sa}}^m$ with density proportional to $\exp (-n^2(\Vert x{\Vert }_2^2/2+f(x)))$. If $f$ is $L$-Lipschitz with respect to $\Vert \cdot {\Vert }_1$, we construct a coupling of $X\sim {\mu }_f$ and an $m$-tuple $Z$ of independent GUE matrices such that $\Vert X-Z{\Vert }_{\infty }\le L$ almost surely, where $\Vert \cdot {\Vert }_{\infty }$ 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.

arXiv comment14 pages