Dynamical clusters of infinite particle dynamics

TL;DR

This paper investigates the structure of interaction graphs in infinite particle systems, providing exponential bounds on their connected components, which advances understanding of percolation in complex geometrical settings.

Contribution

It introduces a detailed analysis of interaction graphs for infinite particle dynamics and establishes exponential estimates for their finite connected components.

Findings

Exponential bounds on the size of connected components in the interaction graph.

Solution to the continuous percolation problem for tubes around particle trajectories.

Enhanced understanding of geometrical percolation in infinite particle systems.

Abstract

For any system $\{i\}$ of particles with the trajectories $x_{i}(t)$ in $R^{d}$ on a finite time interval $[0,\tau]$ we define the interaction graph $G$. Vertices of $G$ are the particles, there is an edge between two particles $i,j$ iff for some $t\in[0,\tau]$ the distance between particles $i,j$ is not greater than some constant. We undertake a detailed study of this graph for infinite particle dynamics and prove exponential estimates for its finite connected components. This solves continuous percolation problem for a complicated geometrical objects - the tubes around particle trajectories.