Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes

Cafasso, Mattia;Claeys, Tom;Girotti, Manuela
(2019) International Mathematics Research Notices — Vol. rnz168, n° rnz168, p. rnz168 (2019)

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  • Cafasso, Mattia
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  • Claeys, Tomorcid-logoUCLouvain
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  • Girotti, Manuela
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Abstract
We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr. In addition, we obtain asymptotics at ±∞ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.
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Cafasso, M., Claeys, T., & Girotti, M. (2019). Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes. International Mathematics Research Notices, rnz168(rnz168), rnz168. https://doi.org/10.1093/imrn/rnz168 (Original work published 2019)