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)
Random matrices, non-colliding processes and queues. (English) Zbl 1041.15019
Azéma, Jacques (ed.) et al., 36th seminar on probability. Berlin: Springer (ISBN 3-540-00072-0/pbk). Lect. Notes Math. 1801, 165-182 (2003).

It was discovered recently by J. Baik, P. Deift and K. Johansson, [J. Am. Math. Soc. 12, No. 4, 1119–1178 (1999; Zbl 0932.05001)] that the asymptotic distribution of the length of the longest increasing subsequence in a permutation chosen uniformly at random from a process ${S}_{n}$, properly centered and normalised, is the same as the asymptotic distribution of the largest eigenvalue of an $n×n$ GUE random matrix, properly centered and normalised, as $n\to \infty$. On the other hand, the eigenvalues of a GUE random matrix of dimension $n$ obey the same law as positions, after a unit length of time, of $n$ independent standard Brownian motions started from the origin and conditioned never to collide. These connections between random matrices, non-colliding processes and queues are studied.

The known theorem of classical queueing theory, which states that, in equilibrium, the output of a stable $M/M/1$ queue is Poisson, was firstly proved by P. J. Burke [Operations Research 4, No. 6, 699–704 (1956)]. Here, this output theorem for the $M/M/1$ queue is proved using reversibility and its extended version is used to obtain a representation for non-colliding Poisson processes and for non-colliding Brownian motions. Finally, output theorems are discussed generally and the analogue of the output theorem is given for Brownian motions with drift together with the first order asymptotics for directed percolation and directed polymers.

MSC:
 15A52 Random matrices (MSC2000) 82D60 Polymers (statistical mechanics) 15A18 Eigenvalues, singular values, and eigenvectors 60K25 Queueing theory 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J65 Brownian motion