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Non-Colliding Random Walks, Tandem Queues, and DiscreteOrthogonal Polynomial Ensembles

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1.	Title	Title of document	Non-Colliding Random Walks, Tandem Queues, and DiscreteOrthogonal Polynomial Ensembles
2.	Creator	Author's name, affiliation, country	Wolfgang König; BRIMS, Hewlett-Packard Laboratories
2.	Creator	Author's name, affiliation, country	Neil O'Connell; BRIMS, Hewlett-Packard Laboratories
2.	Creator	Author's name, affiliation, country	Sébastien Roch; École Polytechnique
3.	Subject	Discipline(s)	Mathematics
3.	Subject	Keyword(s)	Non-collidingrandom walks, tandem queues,
3.	Subject	Subject classification	Primary60J45, 60K25, 60K35; secondary 15A52, 05E35, 60J10, 60J27.
4.	Description	Abstract	We show that the function $h(x)=\prod_{i < j}(x_j-x_i)$ is harmonic for any random walk in $R^k$ with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber $W=\{x\colon x_1 < x_2 < \cdots < x_k\}$ onto a point where $h$ vanishes, we define the corresponding Doob $h$-transform. For certain special cases, we show that the marginal distribution of the conditioned process at a fixed time is given by a familiar discrete orthogonal polynomial ensemble. These include the Krawtchouk and Charlier ensembles, where the underlying walks are binomial and Poisson, respectively. We refer to the corresponding conditioned processes in these cases as the Krawtchouk and Charlier processes. In [O'Connell and Yor (2001b)], a representation was obtained for the Charlier process by considering a sequence of $M/M/1$ queues in tandem. We present the analogue of this representation theorem for the Krawtchouk process, by considering a sequence of discrete-time $M/M/1$ queues in tandem. We also present related results for random walks on the circle, and relate a system of non-colliding walks in this case to the discrete analogue of the circular unitary ensemble (CUE).
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2001-10-12
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/104
10.	Identifier	Digital Object Identifier	10.1214/EJP.v7-104
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 7
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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