Abstract
We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among n sticky Brownian motions has Tracy-Widom distributed fluctuations in the large n and large time limit. These results are proved by viewing sticky Brownian motions as a (previously known) limit of the exactly solvable beta random walk in random environment.
Highlights
Introduction and main results1.1 IntroductionFamilies of interacting Brownian motions have been related to random matrix theory in a number of works
We prove that the random lower order corrections to the large deviation principle, which come from the random environment, are Tracy-Widom Gaussian unitary ensemble (GUE) distributed in the large time limit (Theorem 1.15)
Our results can be rephrased to say that as time and the number of particles n are simultaneously sent to infinity, the position of the extremal particle of n uniform sticky Brownian motions has Tracy-Widom GUE distributed fluctuations (Corollary 1.17). We prove these results by viewing uniform sticky Brownian motions as the limit of a discrete exactly solvable model: the beta random walk in random environment (RWRE)
Summary
Families of interacting Brownian motions have been related to random matrix theory in a number of works. In this paper we restrict our attention to a specific one-parameter family of sticky Brownian motions which we will call uniform sticky Brownian motions where the multiparticle interactions are completely determined by the two particle interactions Within this restricted class, we prove a quenched large deviation principle (Theorem 1.13) for the random heat kernel (referred to below as the uniform Howitt-Warren stochastic flow of kernels). Our results can be rephrased to say that as time and the number of particles n are simultaneously sent to infinity, the position of the extremal particle of n uniform sticky Brownian motions has Tracy-Widom GUE distributed fluctuations (Corollary 1.17) We prove these results by viewing uniform sticky Brownian motions as the limit of a discrete exactly solvable model: the beta random walk in random environment (RWRE).
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