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Zbl 1029.34087
Forrester, P.J.; Witte, N.S.
Application of the $\tau$-function theory of Painlevé equations to random matrices: $\text{P}_V$, $\text{P}_{III}$, the LUE, JUE, and CUE. (English)
[J] Commun. Pure Appl. Math. 55, No.6, 679-727 (2002). ISSN 0010-3640

Summary: With $\langle\cdot\rangle$ denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is $\widetilde E_N(I; a,\mu):= \langle \prod^N_{l=1}\chi^{(l)}_{(0,\infty)\setminus I}(\lambda- \lambda_l)^\mu\rangle$ for $I= (0,s)$ and $I= (s,\infty)$, where $\chi^{(l)}_I= 1$ for $\lambda_l\in I$ and $\chi^{(l)}_I= 0$ otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown that $\widetilde E_N(I; a,\mu)$ is a $\tau$-function associated with the Hamiltonian therein, and so can be characterised as the solution to a certain second-order second-degree differential equation, or in terms of the solution to certain difference equations. The cases $\mu= 0$ and $\mu= 2$ are of particular interest as they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue. In the case $I= (s,\infty)$, $\widetilde E_N(I; a,\mu)$ is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard edge and soft edge scaled limits of $\widetilde E_N(I; a,\mu)$. In particular, in the hard edge scaled limit it is shown that the limiting quantity $E^{\text{hard}}((0, s); a,\mu)$ can be evaluated as a $\tau$-function associated with the Hamiltonian in Okamoto's theory of the Painlevé III equation.

MSC 2000:: *34M55 Painlevé and other special equations
37J99 Finite-dimensional Hamiltonian etc. systems
82B31 Stochastic methods in equilibrium statistical mechanics

Keywords: random matrices; Painlevé equations; reflection groups; interacting random processes

Cited in: Zbl 1098.15017 Zbl 1056.15023

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