# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\text{P}}_{\text{VI}}$, the JUE, CyUE, cJUE and scaled limits. (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)