A Representation for Non-Colliding Random Walks

By Neil O'Connell, Marc Yor. 2002

174 citations recorded by the source corpus.

We define a sequence of mappings $\Gamma_k:D_0(R_+)^k\to D_0(R_+)^k$ and prove the following result: Let $N_1,\ldots,N_n$ be the counting functions of independent Poisson processes on $R_+$ with respective intensities $\mu_1 < \mu_2 < \cdots < \mu_n$. The conditional law of $N_1,\ldots,N_n$, given that $$N_1(t)\le\cdots\le N_n(t), \mbox{ for all }t\ge 0,$$ is the same as the unconditional law of $\Gamma_n(N)$. From this, we deduce the corresponding results for independent Poisson processes of equal rates and for independent Brownian motions (in both of these cases the conditioning is in the sense of Doob). This extends a recent observation, independently due to Baryshnikov (2001) and Gravner, Tracy and Widom (2001), which relates the law of a certain functional of Brownian motion to that of the largest eigenvalue of a GUE random matrix. Our main result can also be regarded as a generalisation of Pitman's representation for the 3-dimensional Bessel process.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e216eda5c811ec64638a45bb583b88b6f9893ebf

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Citations in the corpus

Directed polymers and the quantum Toda lattice (2009, background): “But we do not need these facts, and refer the reader to the papers [6, 40, 43] for details.”
Paths in Weyl chambers and random matrices (2002, methodology): “By making use of the representation of rad(Z) as a conditionned Brownian motion, O’Connell & Yor [ 27 ] have proposed a completely different approach of the study of the path representation of the G.U.E., starting from the Burke reversibility property of queues.”
A GUE central limit theorem and universality of directed first andlast passage site percolation (2004, background): “By studying a special random variable, Baryshnikov [2] and Gravner, Tracy and Widom [8] (see also [14, 13]) showed that (17) P(D̂k(1) ≤ s) = 1 Zk ∫ s −∞ .”
Distribution Functions for Largest Eigenvalues and Their Applications (2002, background): “s shown in [8, 16] that Dˆ k is equal in distribution to the largest eigenvalue of a k × k GUE random matrix. This fascinating connection has been greatly clariﬁed in recent work of O’Connell and Yor [35] (see also [36]). From Johansson [22] it follows for V Poisson that P D(⌊xn⌋,n) −c1n c2 n1/3 &l”
Random matrices, non-colliding processes and queues (2002, unknown): “The proof of this identity in law, given in [47], is based on an extension of the output theorem for a stable M / M /1 queue in equilibrium, and certain related symmetries in the max-plus algebra.”
Dyson's Brownian motions, intertwining and interlacing (2005, methodology): “This is essentially the same argument for (2) as given by O’Connell and Yor, [21], with Proposition 6 replacing the corresponding statement about Γ.”
Littelmann paths and Brownian paths (2004, unknown): “R T ] 11”
A path-transformation for random walks and the Robinson-Schensted correspondence (2002, methodology): “As discussed in [ 36 ], the results for Brownian motion have an interpretation in random matrix theory.”
Combinatorial Stochastic Processes (2006, methodology): “A multidimensional extension is presented in [332], where a representation for Brownian motion conditioned never to exit a (type A) Weyl chamber is obtained using reversibility and symmetry properties of certain queueing networks.”
A Universality Property for Last-Passage Percolation Paths Close to the Axis (2004, unknown): “The random variable L (1 , k ) has the same distribution as the largest eigenvalue of a k × k GUE random matrix [4], [7], [15].”