Computing extreme singular values of free operators

TL;DR

This work develops explicit, tractable variational formulas for the extreme spectral edges of non-self-adjoint random matrices modeled by free probabilistic constructions. By linearizing and dilating to a free Fock-space framework, the authors extend Lehner's edge formulas to the non-self-adjoint setting, deriving sharp expressions for $\lambda_{\max}(xx^* + b\otimes\mathbf{1})$ and $\lambda_{\min}(xx^* + b\otimes\mathbf{1})$ as optimized matrix-functionals over positive/negative semidefinite variables $z$. A reduction principle exploiting algebraic symmetries simplifies these variational problems to lower-dimensional, matrix-valued or diagonal forms in common applications, and a matrix-valued Cauchy transform identity provides a unifying structural tool. The results yield matrix-valued analogues of the free Poisson distribution and enable practical computation of spectral edges for a broad class of non-homogeneous random matrices through deterministic free-probability computations.

Abstract

A recent development in random matrix theory, the intrinsic freeness principle, establishes that the spectrum of very general random matrices behaves as that of an associated free operator. This reduces the study of such random matrices to the deterministic problem of computing spectral statistics of the free operator. In the self-adjoint case, the spectral edges of the free operator can be computed exactly by means of a variational formula due to Lehner. In this note, we provide variational formulas for the largest and smallest singular values in the non-self-adjoint case.

Theorems & Definitions (38)

• Theorem 1.1: Lehner
• Theorem 1.2
• Corollary 1.3
• proof
• Lemma 1.4
• Corollary 1.5
• proof
• Lemma 2.1: Schur complements
• Lemma 2.2: Matrix inversion lemma
• proof : Proof of Theorem \ref{['thm:lehner']}