Local Statistics of Singular Values for Products of Truncated Unitary Matrices

TL;DR

This paper studies the local spectral statistics of singular values in products of large truncated unitary matrices, revealing a universal three-phase transition between GUE and Gaussian statistics, extending previous results from Gaussian matrices.

Contribution

It provides the first complete characterization of the three-phase transition in spectral statistics for products of truncated unitary matrices, generalizing known results from Gaussian matrices.

Findings

Identifies a universal three-phase transition in spectral statistics.

Demonstrates interpolation between GUE and Gaussian behaviors.

Extends spectral transition results to truncated unitary matrices.

Abstract

This paper investigates local spectral statistics of singular values for many products of independent large rectangular matrices, sampled from the ensemble of truncated unitary matrices with the invariant Haar measure. Our main contribution establishes a universal three-phase transition in these statistics, demonstrating an interpolation beween GUE statistics and classical Gaussian behavior. While such transition was previously known for products of complex Gaussian matrices\cite{ABK19}\cite{LWW23}, the current work provides the complete characterization in the truncated unitary matrix setting.