Sum rules and large deviations for spectral measures on the unit circle

2016/04/23 by Fabrice Gamboa, Gamboa, Fabrice, Jan Nagel +3 · 2 citations
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.1604.06934

openalex publication_date 2016/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work is a companion paper of Gamboa, Nagel, Rouault (J. Funct. Anal. 2016). We continue to explore the connections between large deviations for random objects issued from random matrix theory and sum rules. Here, we are concerned essentially with measures on the unit circle whose support is an arc that is possibly proper. We particularly focus on two matrix models. The first one is the Gross-Witten ensemble. In the gapped regime we give a probabilistic interpretation of a Simon sum rule. The second matrix model is the Hua-Pickrell ensemble. Unlike the Gross-Witten ensemble the potential is here infinite at one point. Surprisingly, but as in the above mentioned paper, we obtain a completely new sum rule for the deviation to the equilibrium measure of the Hua-Pickrell ensemble. The extension to matrix measures is also studied.

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• Sum rules via large deviations: polynomial potentials and multi-cut regime on the unit circle
• Matrix-Driven Identification and Reconstruction of LLM Weight Homology

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