  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. These systems are characterized by a parameter that controls
the strength of the interaction, $k>0$. In spite of being a one-dimensional
system, this system is interesting due to the properties that arise from the
long-range interaction between particles. It is a well-known fact that when $k$
is small enough, particle 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, namely, the origin.
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.
