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July 2013 Nonintersecting random walks in the neighborhood of a symmetric tacnode
Mark Adler, Patrik L. Ferrari, Pierre van Moerbeke
Ann. Probab. 41(4): 2599-2647 (July 2013). DOI: 10.1214/11-AOP726

Abstract

Consider a continuous time random walk in $\mathbb{Z} $ with independent and exponentially distributed jumps $\pm1$. The model in this paper consists in an infinite number of such random walks starting from the complement of $\{-m,-m+1,\ldots,m-1,m\}$ at time $-t$, returning to the same starting positions at time $t$, and conditioned not to intersect. This yields a determinantal process, whose gap probabilities are given by the Fredholm determinant of a kernel. Thus this model consists of two groups of random walks, which are contained within two ellipses which, with the choice $m\simeq2t$ to leading order, just touch: so we have a tacnode. We determine the new limit extended kernel under the scaling $m=\lfloor2t+\sigma t^{1/3}\rfloor$, where parameter $\sigma$ controls the strength of interaction between the two groups of random walkers.

Citation

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Mark Adler. Patrik L. Ferrari. Pierre van Moerbeke. "Nonintersecting random walks in the neighborhood of a symmetric tacnode." Ann. Probab. 41 (4) 2599 - 2647, July 2013. https://doi.org/10.1214/11-AOP726

Information

Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1279.60062
MathSciNet: MR3112926
Digital Object Identifier: 10.1214/11-AOP726

Subjects:
Primary: 35Q53, 60G55, 60G60
Secondary: 35Q58, 60G10

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 4 • July 2013
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