Dynamical clusters of infinite particle dynamics

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.