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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\text{P}}_{V}$, ${\text{P}}_{III}$, the LUE, JUE, and CUE. (English) Zbl 1029.34087
Summary: With $〈·〉$ denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is ${\stackrel{˜}{E}}_{N}\left(I;a,\mu \right):=〈{\prod }_{l=1}^{N}{\chi }_{\left(0,\infty \right)\setminus I}^{\left(l\right)}{\left(\lambda -{\lambda }_{l}\right)}^{\mu }〉$ for $I=\left(0,s\right)$ and $I=\left(s,\infty \right)$, where ${\chi }_{I}^{\left(l\right)}=1$ for ${\lambda }_{l}\in I$ and ${\chi }_{I}^{\left(l\right)}=0$ otherwise. Using Okamoto’s development of the theory of the Painlevé V equation, it is shown that ${\stackrel{˜}{E}}_{N}\left(I;a,\mu \right)$ 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 $I=\left(s,\infty \right)$, ${\stackrel{˜}{E}}_{N}\left(I;a,\mu \right)$ 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 ${\stackrel{˜}{E}}_{N}\left(I;a,\mu \right)$. In particular, in the hard edge scaled limit it is shown that the limiting quantity ${E}^{\text{hard}}\left(\left(0,s\right);a,\mu \right)$ can be evaluated as a $\tau$-function associated with the Hamiltonian in Okamoto’s theory of the Painlevé III equation.

##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 82B31 Stochastic methods in equilibrium statistical mechanics