Gap Probability of the Circular Unitary Ensemble with a Fisher–Hartwig Singularity and the Coupled Painlevé V System

Abstract

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.

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Acknowledgements

The work of Shuai-Xia Xu was supported in part by the National Natural Science Foundation of China under Grant Numbers 11571376 and 11971492. Yu-Qiu Zhao was supported in part by the National Natural Science Foundation of China under Grant Numbers 11571375 and 11971489.

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Xu, S., Zhao, Y. Gap Probability of the Circular Unitary Ensemble with a Fisher–Hartwig Singularity and the Coupled Painlevé V System. Commun. Math. Phys. 377, 1545–1596 (2020). https://doi.org/10.1007/s00220-020-03776-3

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