Non-intersecting random walks in the neighborhood of a symmetric tacnode

Details Non-intersecting random walks in the neighborhood of a symmetric tacnode Non-intersecting random walks in the neighborhood of a symmetric tacnode

2010-07-07

arxiv.org/abs/1007.1163v2

Physics

Mathematical Physics

50 pages, 5 figures, LaTeX; We extended the previous result from one- to multi-time and improved the presentation too

Scientific paper

Consider a continuous time random walk in Z with independent and exponentially distributed jumps +/-1. The model in this paper consists in an infinite number of such random walks starting from the complement of {-m,-m+1,...,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 into two ellipses which, with the choice m=2t to leading order, just touch: so we have a tacnode. We determine the new limit extended kernel under the scaling m=[2t+sigma t^{1/3}], where parameter sigma controls the strength of interaction between the two groups of random walkers.

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