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We consider systems of multiple Brownian particles in one dimension thatrepel mutually via a logarithmic potential on the real line, more specificallythe Dyson model. These systems are characterized by a parameter that controlsthe strength of the interaction, $k>0$. In spite of being a one-dimensionalsystem, this system is interesting due to the properties that arise from thelong-range interaction between particles. It is a well-known fact that when $k$is small enough, particle collisions occur almost surely, while when $k$ islarge, collisions never occur. However, aside from this fact there was nocharacterization of the collision times until now. In this paper, we derive thefractal (Hausdorff) dimension of the set of collision times by generalizingtechniques introduced by Liu and Xiao to study the return times to the originof self-similar stochastic processes. In our case, we consider the return timesto configurations where at least one collision occurs, which is a conditionthat defines unbounded sets, as opposed to a single point, namely, the origin.We find that the fractal dimension characterizes the collision behavior ofthese systems, and establishes a clear delimitation between the colliding andthe non-colliding regions in a way similar to that of an order parameter.

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