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

Abstract

Let $A_0,A_1,\dots ,A_n\in M_N(\mathbb{C})$ be Hermitian matrices and let $G_1,\dots ,G_n$ be independent $M\times M$ GUE matrices normalized so that $\Vert M^{-1/2}{G}_i\Vert \to 2$ 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 ${\rho }_{+}={\inf }_{Z\succ 0}{\lambda }_{\max }(A_0+Z+{\sum }_{i=1}^n{A}_i{Z}^{-1}{A}_i)$ and ${\rho }_{-}={\sup }_{Z\prec 0}{\lambda }_{\min }(A_0+Z+{\sum }_{i=1}^n{A}_i{Z}^{-1}{A}_i)$. Assuming $A_i⪰0$ for $i\ge 1$ and $M\ge N$, we prove via concentration and minimax duality the finite-dimensional bounds $\mathbb{E}{\lambda }_{\max }(H_M)\le {\rho }_{+}+9\sqrt{nN/M}\Vert {\sum }_{i=1}^n{A}_i^2{\Vert }_{\mathrm{op}}^{1/2}$ and $\mathbb{E}{\lambda }_{\min }(H_M)\ge {\rho }_{-}-9\sqrt{nN/M}\Vert {\sum }_{i=1}^n{A}_i^2{\Vert }_{\mathrm{op}}^{1/2}$. With ${\rho }_{\ast }=\max \{{\rho }_{+},-{\rho }_{-}\}$, this yields $\mathbb{E}\Vert H_M{\Vert }_{\mathrm{op}}\le {\rho }_{\ast }+9\sqrt{nN/M}\Vert {\sum }_{i=1}^n{A}_i^2{\Vert }_{\mathrm{op}}^{1/2}$. For uniformly bounded positive coefficients, bounded $n$, and $N=o(M)$, one obtains ${\mathrm{limsup}}_{M\to \infty }\mathbb{E}\Vert H_M{\Vert }_{\mathrm{op}}\le \rho$ whenever ${\rho }_{\ast ,M}\to \rho$. 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.

17 pages