On an Average over the Gaussian Unitary Ensemble

2009/03/31 by Francesco Mezzadri, F. Mezzadri, M. Y. Mo · 1 citation
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #Stochastic processes and statistical mechanics #math-ph #math.MP #msc:15A52 #msc:35Q15

paper · pdf · doi:10.1093/imrn/rnp062

published as Int. Math. Res. Notices (2009), Vol. 2009: article ID rnp062, 30 pages · 28 pages, 3 figures

arxiv created 2009/03/31 · openalex publication_date 2009/05/24 · arxiv updated 2010/04/13 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic limit for large matrix dimension N of the partition function of the unitary ensemble (β = 2) with weight ⁠. We compute the leading-order term of the partition function and the coefficients of its Taylor expansion. Our results are valid in the region ⁠. Such a partition function contains all the information on a new statistics of the eigenvalues of matrices in the Gaussian unitary ensemble that was introduced by Berry and Shukla [2]. It can also be interpreted as the moment-generating function of the singular linear statistics ⁠.

Citations

• Tuck's incompressibility function: statistics for zeta zeros and eigenvalues
• Painléve III and a singular linear statistics in Hermitian random matrix ensembles I

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• Singular-potential random-matrix model arising in mean-field glassy systems