Local Statistics of Singular Values for Products of Truncated Unitary Matrices

2025/06/17 by Y. T. Gu, Gu, Yandong, Dang-Zheng Liu +1 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #60B20 #Advanced Mathematical Theories and Applications #FOS: Mathematics #Matrix Theory and Algorithms #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2506.14193

openalex publication_date 2025/06/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

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\citeABK19\citeLWW23, the current work provides the complete characterization in the truncated unitary matrix setting.

Citations

• Last-passage percolation and product-matrix ensembles
• Global Universality of Singular Values in Products of Many Large Random Matrices
• Lyapunov exponents for truncated unitary and Ginibre matrices
• Non-asymptotic Results for Singular Values of Gaussian Matrix Products
• Phase transitions for infinite products of large non-Hermitian random matrices
• Gaussian fluctuations for products of random matrices
• Lyapunov exponent, universality and phase transition for products of random matrices
• Spectral Densities of Singular Values of Products of Gaussian and\n Truncated Unitary Random Matrices
• Bulk and soft-edge universality for singular values of products of Ginibre random matrices
• Spectral Radii of Large Non-Hermitian Random Matrices
• Plancherel-Rotach formulae for average characteristic polynomials of\n products of Ginibre random matrices and the Fuss-Catalan distribution
• On the asymptotic distribution of the singular values of powers of random matrices

Cited by

• Global and local limits for products of rectangular Ginibre matrices
• Edge statistics for singular values of products of rectangular complex Ginibre matrices

Related

• Random matrix products: Universality and least singular values
• Global and local limits for products of rectangular Ginibre matrices
• Spectral Densities of Singular Values of Products of Gaussian and\n Truncated Unitary Random Matrices
• Spectral statistics for product matrix ensembles of Hermite type with external source
• Bulk and soft-edge universality for singular values of products of Ginibre random matrices