Collision types and times in interacting particle systems

Abstract

We consider a system of stochastic interacting particles with general diffusion coefficient and drift functions and we study the types of collisions that arise in them. In particular, interactions between particles are inversely proportional to their separation, and the coupling function of interaction is also considered in great generality. Our main result indicates that under very mild conditions, all collisions are simple almost surely, namely, only one pair of particles collides at any time, while more complicated collisions such as three-body or disjoint two-body collisions occur with zero probability. In order to obtain our results we make use of symmetric polynomials on the square of particle separations; the degree of these polynomials indicates the type of collision, and by a locality argument we show that polynomials indicating a non-simple collision almost surely do not cancel. We make use of our main result to study the Hausdorff dimension of times at which collisions occur, and we show that this dimension is given by the ratio between the interaction coupling and diffusion functions. Our results cover many of the most well-known particle systems, such as the Dyson model and Wishart processes and their extensions to non-constant diffusion coefficients and background drifts.