Non-Colliding Random Walks, Tandem Queues, and Discrete Orthogonal Polynomial Ensembles
Wolfgang König, BRIMS, Hewlett-Packard Laboratories
Neil O'Connell, BRIMS, Hewlett-Packard Laboratories
Sébastien Roch, École Polytechnique
Abstract
We show that the function h(x)=prodi < j(xj-xi)
is harmonic for any random walk in Rk with exchangeable
increments, provided the required moments exist. For the
subclass of random walks which can only exit the Weyl chamber
W={x: x1 < x2 < ...
< xk} 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).
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