Application of the τ-Function Theory¶of Painlevé Equations to Random Matrices:¶PIV, PII and the GUE
By Peter J. Forrester, Nicholas S. Witte. 2001
216 citations recorded by the source corpus.
Abstract: Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of , where for and otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of . Of particular interest are and FN(λ;2), and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto τ-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities and FN(λ;a) for the other classical matrix ensembles.
Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/0a6a66e2de41f998c25694c74ea990b1f9ba5867
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In Condensed Matter Physics
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Application of the τ-function theory of Painlevé equations to random matrices: PVI , the JUE, CyUE, cJUE and scaled limits (2002, background): “A fundamental fact is that as a function of s, and after multiplication by a suitable elementary function of s, these functions satisfy a classical second order linear differential equation (Hermite-Weber equation in the Gaussian case [29], confluent hypergeometric equation in the Laguerre case [30]”