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 $\tau$-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. (English) Zbl 1042.82019
Summary: {\it C. A. Tracy} and {\it H. Widom} [Commun. Math. Phys. 159, 151--174 (1994; Zbl 0789.35152))] have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PTV and PII transcendent respectively. We generalise these results to the evaluation of $$\widetilde E_N(\lambda:a): =\left\langle \prod^N_{l=1} \chi^{(l)}_{(-\infty, \lambda]} (\lambda-\lambda_l)^a \right\rangle,$$ where $\chi^{(l)}_{(-\infty, \lambda]} =1$ for $\lambda_l\in(-\infty,\lambda]$ and $\chi^{(l)}_{(-\infty, \lambda]}=0$ otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of $$F_N(\lambda;a): = \left\langle \prod^N_{l=1}(\lambda-\lambda_l)^a \right\rangle.$$ Of particular interest are $\widetilde E_N(\lambda;2)$ and $F_N(\lambda;2)$, and their scaled limits, which gives the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto $\tau$-function theory of PIV and PII [cf, {\it K. Okamoto}, [Studies on the Painlevé Equations. III. Second and Fourth Painlevé Equations, $P_{II}$ and $P_{IV}$. Math. Ann. 275, 221--255 (1986; Zbl 0589.58008)], for which we give a self contained presentation based on the recent work of {\it M. Noumi} and {\it Y. Yamada} [Nagoya Math. J. 153, 53--86 (1999; Zbl 0932.34088)]. We point out that the same approach can be used to study the quantities $\widetilde E_N(\lambda;a)$ and $F_N(\lambda;a)$ for the other classical matrix ensembles.

82B41Random walks, random surfaces, lattice animals, etc. (statistical mechanics)
15B52Random matrices
34M55Painlevé and other special equations; classification, hierarchies
Full Text: DOI arXiv