Eigenvalue Statistics for CMV Matrices: From Poisson to Clock via Circular Beta Ensembles

2006/07/31 by Rowan Killip, Killip, Rowan, Mihai Stoiciu +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #34B05 #42C05 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.PR #msc:34B05 #msc:42C05

paper · pdf · doi:10.48550/arxiv.math-ph/0608002

arxiv created 2006/07/31 · openalex publication_date 2006/07/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study CMV matrices (a discrete one-dimensional Dirac-type operator) with random decaying coefficients. Under mild assumptions we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay we obtain the circular beta ensembles of random matrix theory. The temperature β^-1 appears as the square of the coupling constant.

Citations

• Interpretations of some parameter dependent generalizations of classical matrix ensembles

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