*(English)*Zbl 1056.15023

This paper is the last of a series [Commun. Pure Appl. Math. 55, No. 6, 679–727 (2002; Zbl 1029.34087) and Commun. Math. Phys. 219, No. 2, 357–398 (2001; Zbl 1042.82019)] where the authors explore the applications of the Okamoto $\tau $-function of Painlevé equations to the characterization of certain averages in random matrix theory.

In this work the authors focus in the Jacobi unitary ensemble (JUE) and the Cauchy unitary ensemble (CyUE). Some averages over the eigenvalue probability density function for the JUE and the CyUE are identified with some multi-dimensional integrals, and these integrals are identified with the determinant of integral solutions of the Gauss hypergeometric equation. Thus those averages are characterized as the solution of the second order second degree equation satisfied by the Hamiltonian in the ${P}_{\phantom{\rule{4.pt}{0ex}}\text{VI}}$ theory.

Many applications to random matrix theory are provided. Namely: to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart; to the expression for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE); to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the $\tau -$function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter $a=0$; and to the characterization of the diagonal-diagonal spin-spin correlation in the two dimensional Ising model.

##### MSC:

15A52 | Random matrices (MSC2000) |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

33E17 | Painlevé-type functions |

34M55 | Painlevé and other special equations; classification, hierarchies |

34A05 | Methods of solution of ODE |

15A18 | Eigenvalues, singular values, and eigenvectors |

82B43 | Percolation (equilibrium statistical mechanics) |