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arXiv · 1007.1163

Nonintersecting random walks in the neighborhood of a symmetric tacnode

Mark Adler·Patrik L. Ferrari·Pierre van Moerbeke
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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.

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Mark Adler, Patrik L. Ferrari, Pierre van Moerbeke. 2010-07-07. Nonintersecting random walks in the neighborhood of a symmetric tacnode. https://doi.org/10.1214/11-aop726

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