On sub-graphs of the Johnson $\mathcal{J}(n,k)$ graph induced by particle systems on simple graphs

TL;DR

This paper investigates the topology of state spaces for particle systems on simple graphs, showing they can be embedded into Johnson graphs under certain movement and exclusion constraints, providing new insights into their structure.

Contribution

It introduces a framework for analyzing particle systems with exclusion on simple graphs by embedding their state spaces into Johnson graphs, addressing combinatorial challenges.

Findings

Particle systems with exclusion can be embedded into Johnson graphs.

The topology of state spaces is characterized for systems with single-particle moves.

Provides tools for analyzing complex particle interactions on graphs.

Abstract

Understanding the topology of the state space has proven to be extremely efficient for dynamical systems with a continuous state space. On the other hand, for particle systems on finite simple graphs, it has not yet been subject to deep investigation due to combinatorial hurdles and existing efficient spectral theoretic approaches to the analysis of classical particle systems. In the context of complex systems with heterogeneous interactions of particles, these techniques can break down due to intractability. This work provides a tool box of results on the topology of state spaces of particles systems under the sole conditions that exactly one particle moves at a time and no two particles may occupy the same vertex at the same time. The Johnson $\mathcal{J}(n,k)$ graph yields the overarching structure and we prove that any particle system with exclusion property and single particle movements may be embedded as a dynamical system on a sub-graph of $\mathcal{J}(n,k)$.