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

TL;DR

This paper investigates the fractal Hausdorff dimension of collision times in one-dimensional Dyson log-gases, revealing how it characterizes the transition between colliding and non-colliding particle regimes based on interaction strength.

Contribution

It provides the first characterization of the Hausdorff dimension of collision times in Dyson log-gases, extending techniques to analyze return times to collision configurations.

Findings

Hausdorff dimension distinguishes colliding and non-colliding regimes

Dimension varies with interaction strength parameter

Characterizes the set of collision times as a fractal set

Abstract

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.