zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 τ-function theory of Painlevé equations to random matrices: P V , 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 E ˜ N (I;a,μ):= l=1 N χ (0,)I (l) (λ-λ l ) μ for I=(0,s) and I=(s,), where χ I (l) =1 for λ l I and χ I (l) =0 otherwise. Using Okamoto’s development of the theory of the Painlevé V equation, it is shown that E ˜ N (I;a,μ) is a τ-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 μ=0 and μ=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,), E ˜ N (I;a,μ) 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 E ˜ N (I;a,μ). In particular, in the hard edge scaled limit it is shown that the limiting quantity E hard ((0,s);a,μ) can be evaluated as a τ-function associated with the Hamiltonian in Okamoto’s theory of the Painlevé III equation.

34M55Painlevé and other special equations; classification, hierarchies
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
82B31Stochastic methods in equilibrium statistical mechanics