2019/07/31 by Shuai-Xia Xu, Yu-Qiu Zhao · 1 citation
Mathematics · Physics and Astronomy · #Eigenvalues and eigenvectors #Essential singularity #Hamiltonian (control theory) #Mathematical functions and polynomials #Microcanonical ensemble #Probability measure #Random Matrices and Applications #Random matrix #Singularity #Spectral Theory in Mathematical Physics #Toeplitz matrix #Unit circle #math-ph #math.MP #msc:33E17 #msc:34M55 #msc:41A60
paper · pdf · doi:10.1007/s00220-020-03776-3
52 pages, 6 figures, misprints corrected
openalex created_date 2019/08/13 · openalex publication_date 2020/06/05 · arxiv created 2020/06/07 · arxiv updated 2020/06/09 · openalex updated_date 2026/08/05
We consider the circular unitary ensemble with a Fisher-Hartwig singularity of both jump type and root type at z=1. A rescaling of the ensemble at the Fisher-Hartwig singularity leads to the confluent hypergeometric kernel. By studying the asymptotics of the Toeplitz determinants, we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the Hamiltonian of the coupled Painlevé V system in dimension four. This leads to a Painlevé-type representation of the confluent hypergeometric-kernel determinant. Moreover, the large gap asymptotics, including the constant terms, are derived by evaluating the total integral of the Hamiltonian. In particular, we reproduce the large gap asymptotics of the confluent hypergeometric-kernel determinant obtained by Deift, Krasovsky and Vasilevska, and the sine-kernel determinant as a special case, including the constant term conjectured earlier by Dyson.