Dynamics at the edge for independent diffusing particles

TL;DR

This paper investigates the behavior of the extreme particles in a large system of independent Brownian particles, deriving joint distributions and counting statistics at the edge to understand their dynamics over multiple times.

Contribution

It introduces new methods to derive multi-time joint distributions of the maximum and second maximum positions for independent Brownian particles, advancing understanding of edge dynamics.

Findings

Derived multi-time joint distribution of the rightmost particle

Obtained joint distribution of maximum and second maximum positions

Analyzed counting statistics and arrival times at multiple points

Abstract

We study the dynamics of the outliers for a large number of independent Brownian particles in one dimension. We derive the multi-time joint distribution of the position of the rightmost particle, by two different methods. We obtain the two time joint distribution of the maximum and second maximum positions, and study the counting statistics at the edge. Finally we derive the multi-time joint distribution of the running maximum, as well as the joint distribution of the arrival time of the first particle at several space points.