Stationary coalescing walks on the lattice II: Entropy

TL;DR

This paper investigates the entropic properties of coalescing walks on integer lattices, establishing conditions under which bi-infinite trajectories carry entropy and constructing examples with bi-infinite entropy-carrying trajectories.

Contribution

It provides new theoretical results linking entropy to the structure of coalescing walks and constructs explicit models demonstrating these properties.

Findings

Bi-infinite trajectories in systems with completely positive entropy must carry entropy.

Positive entropy in 2D directed walks prevents all trajectories from being bi-infinite.

Constructed a symmetric exclusion process with bi-infinite entropy-carrying trajectories.

Abstract

This paper is a sequel to Chaika and Krishnan [arXiv:1612.00434]. We again consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice Z^d. We assume that once walks meet, they coalesce. We consider various entropic properties of these systems. We show that in systems with completely positive entropy, bi-infinite trajectories must carry entropy. In the case of directed walks in dimension 2 we show that positive entropy guarantees that all trajectories cannot be bi-infinite. To show that our theorems are proper, we construct a stationary discrete-time symmetric exclusion process whose particle trajectories form bi-infinite trajectories carrying entropy.