Nonintersecting random walks in the neighborhood of a symmetric tacnode

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 within 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 +σt^1/3], where parameter σ controls the strength of interaction between the two groups of random walkers. © Institute of Mathematical Statistics, 2013.