A tail bound for multi-matrix models with Lipschitz interaction

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