A Sudakov--Fernique proof of Lehner-type edge bounds for matrix-valued GUE sums

Abstract

Let $A_0,A_1,\ldots,A_n\in M_N(\mathbb{C})$ be Hermitian matrices and let $G_1,\ldots,G_n$ be independent $M\times M$ GUE matrices normalized so that $\|M^{-1/2}G_i\|\to 2$ almost surely as $M\to\infty$. We study the spectral edges and operator norm of $H_M = A_0\otimes I_M + \frac{1}{\sqrt{M}}\sum_{i=1}^n A_i\otimes G_i$. Lehner's formula identifies the right and left edges of the corresponding free semicircular operator as $\rho_+ = \inf_{Z\succ 0}\lambda_{\max}(A_0+Z+\sum_{i=1}^n A_iZ^{-1}A_i)$ and $\rho_- = \sup_{Z\prec 0}\lambda_{\min}(A_0+Z+\sum_{i=1}^n A_iZ^{-1}A_i)$. Assuming $A_i\succeq 0$ for $i\ge 1$ and $M\ge N$, we prove via concentration and minimax duality the finite-dimensional bounds $\mathbb{E}\lambda_{\max}(H_M)\le \rho_+ + 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$ and $\mathbb{E}\lambda_{\min}(H_M)\ge \rho_- - 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$. With $\rho_* = \max\{\rho_+,-\rho_-\}$, this yields $\mathbb{E}\|H_M\|_{\mathrm{op}}\le \rho_* + 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$. For uniformly bounded positive coefficients, bounded $n$, and $N=o(M)$, one obtains $\limsup_{M\to\infty}\mathbb{E}\|H_M\|_{\mathrm{op}}\le\rho$ whenever $\rho_{*,M}\to\rho$. The proof is a matrix-coefficient extension of classical Sudakov--Fernique comparison, combined with a Davidson--Szarek-type singular-value estimate and dual variational formulas for Lehner's edge quantities over density matrices. We also explain why this approach does not extend sharply to signed Hermitian coefficients.