Higher Order Analogues of Tracy-Widom Distributions via the Lax Method

Akemann G, Atkin M (2013)
J.Phys. A 46(1).

Journal Article | Published | English

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We study the distribution of the largest eigenvalue in formal Hermitianone-matrix models at multicriticality, where the spectral density acquires anextra number of k-1 zeros at the edge. The distributions are directly expressedthrough the norms of orthogonal polynomials on a semi-infinite interval, as analternative to using Fredholm determinants. They satisfy non-linear recurrencerelations which we show form a Lax pair, making contact to the stringliterature in the early 1990's. The technique of pseudo-differential operatorsallows us to give compact expressions for the logarithm of the gap probabilityin terms of the Painleve XXXIV hierarchy. These are the higher order analoguesof the Tracy-Widom distribution which has k=1. Using known Backlundtransformations we show how to simplify earlier equivalent results that arederived from Fredholm determinant theory, valid for even k in terms of thePainleve II hierarchy.
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Akemann G, Atkin M. Higher Order Analogues of Tracy-Widom Distributions via the Lax Method. J.Phys. A. 2013;46(1).
Akemann, G., & Atkin, M. (2013). Higher Order Analogues of Tracy-Widom Distributions via the Lax Method. J.Phys. A, 46(1).
Akemann, G., and Atkin, M. (2013). Higher Order Analogues of Tracy-Widom Distributions via the Lax Method. J.Phys. A 46.
Akemann, G., & Atkin, M., 2013. Higher Order Analogues of Tracy-Widom Distributions via the Lax Method. J.Phys. A, 46(1).
G. Akemann and M. Atkin, “Higher Order Analogues of Tracy-Widom Distributions via the Lax Method”, J.Phys. A, vol. 46, 2013.
Akemann, G., Atkin, M.: Higher Order Analogues of Tracy-Widom Distributions via the Lax Method. J.Phys. A. 46, (2013).
Akemann, Gernot, and Atkin, Max. “Higher Order Analogues of Tracy-Widom Distributions via the Lax Method”. J.Phys. A 46.1 (2013).
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