Edge statistics for singular values of products of rectangular complex Ginibre matrices

TL;DR

This paper studies the edge behavior of singular values in products of rectangular complex Ginibre matrices, revealing a phase transition from Gaussian to Airy kernel statistics governed by the depth-to-width ratio.

Contribution

It introduces the DWR as a critical parameter controlling the transition in edge statistics, extending previous work on the critical kernel.

Findings

Gaussian fluctuations in low-DWR regime

Airy kernel at the soft edge in high-DWR regime

DWR acts as a sharp threshold for statistical behavior

Abstract

We investigate the edge statistics of singular values for products of independent rectangular complex Ginibre matrices. Building on work \cite{LWW23}, which introduced the depth-to-width ratio (DWR) and established the critical kernel, we show that the DWR acts as a sharp threshold: local statistics exhibit Gaussian fluctuations in the low-DWR regime, while the Airy kernel emerges at the soft edge in the high-DWR regime.