# The simple exclusion process on finite connected graphs

## TL;DR

This paper analyzes the simple exclusion process on finite connected graphs, identifying invariant measures and showing that occupation time ratios converge to one as particle number approaches vertex count, with explicit calculations for specific graph types.

## Contribution

It characterizes invariant measures and demonstrates the convergence of occupation time ratios in the simple exclusion process on finite graphs, including explicit examples.

## Key findings

- Occupation time at vertex x increases with degree and rate.
- Ratios of occupation times at different vertices converge to one.
- Explicit occupation times computed for star and path graphs.

## Abstract

Consider a system of $K$ particles moving on the vertex set of a finite connected graph with at most one particle per vertex. If there is one, the particle at $x$ chooses one of the $\hbox{deg} (x)$ neighbors of its location uniformly at random at rate $\rho_x$, and jumps to that vertex if and only if it is empty. Using standard probability techniques, we identify the set of invariant measures of this process to study the occupation time at each vertex. Our main result shows that, though the occupation time at vertex $x$ increases with $\hbox{deg} (x) / \rho_x$, the ratio of the occupation times at two different vertices converges monotonically to one as the number of particles increases to the number of vertices. The occupation times are also computed explicitly for simple examples of finite connected graphs: the star and the path.