Hausdorff dimension of collision times in one-dimensional log-gases

We consider systems of multiple Brownian particles in one dimension that repel mutually via a logarithmic potential on the real line, more specifically the Dyson model and the Wishart-Laguerre process. These systems are mainly characterized by a parameter that controls the strength of the interaction, $k>0$. It is a well-known fact that when $k$ is small enough, collisions occur almost surely, while when $k$ is large, collisions never occur. However, aside from this fact there was no characterization of the collision times until now. In this paper, we derive the fractal (Hausdorff) dimension of the set of collision times by generalizing techniques introduced by Liu and Xiao to study the return times to the origin of self-similar stochastic processes. In our case, we consider the return times to configurations where at least one collision occurs, which is a condition that defines unbounded sets, as opposed to a single point. We find that the fractal dimension characterizes the collision behavior of these systems, and establishes a clear delimitation between the colliding and the non-colliding regions in a way similar to that of an order parameter.