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

By W. König, Neil O'Connell, S. Roch. 2002

105 citations recorded by the source corpus.

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).

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1d139f2e714639ee64224898e882ad936d809235

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Citations in the corpus

Random walks in cones (2011, methodology): “This has been shown by König, O’Connell and Roch [34].”
The Arctic circle boundary and the airy process (2003, methodology): “This can be extended to the other parts of the boundary, away from the axes, and to other values of q. This extends the relation between the Meixner ensemble and the Krawtchouk ensemble given in Lemma 2.9 in [15], see also [ 20 ].”
Discrete Polynuclear Growth and Determinantal Processes (2002, unknown): “measure on ZN is given by 1 n! det(K(0,xµ;0,xν)) 1≤µ,ν≤n = 1 n! det( 1 N Xν j=−ν e2πij(xµ−xν)/N) 1≤µ,ν≤n = 1 n!Nn Y 1≤µ<ν≤n |e2πixµ/N −e2πixν/N|2, the equilibrium measure on ZN (discrete CUE), see [23]. We can take the limit n,N→ ∞, n/N→ ρ, 0 <ρ<1, and obtain a limiting determinantal proces”
Combinatorial Stochastic Processes (2006, background): “See also [331, 262] and the survey paper [330].”
Conditional limit theorems for ordered random walks (2009, background): “n, which can be represented as the evolution of k Brownian motions conditioned never to collide. Such conditional versions of random walks have attract a lot of attention in the recent past, see e.g. [14, 11]. The approach in these papers is based on explicit formulasfor nearest-neighbourrandom walk”