On the Widom-Rowlinson Occupancy Fraction in Regular Graphs

TL;DR

This paper establishes a tight upper bound on the occupancy fraction in the Widom-Rowlinson model for d-regular graphs, showing that unions of complete graphs uniquely maximize this bound and related partition functions, confirming a conjecture of Galvin.

Contribution

The paper proves a new tight upper bound on the occupancy fraction in the Widom-Rowlinson model and identifies the extremal graphs as unions of complete graphs, confirming a conjecture of Galvin.

Findings

Unions of complete graphs $K_{d+1}$ uniquely maximize the occupancy fraction.

Maximization of the normalized partition function occurs at $K_{d+1}$.

Confirmed Galvin's conjecture on homomorphism counts for d-regular graphs.

Abstract

We consider the Widom-Rowlinson model of two types of interacting particles on d-regular graphs. We prove a tight upper bound on the occupancy fraction, the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on d+1 vertices, $K_{d+1}$'s. As a corollary we find that $K_{d+1}$ also maximises the normalised partition function of the Widom-Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalised number of homomorphisms from any d-regular graph $G$ to the graph $H_{WR}$, a path on three vertices with a loop on each vertex, is maximised by $K_{d+1}$. This proves a conjecture of Galvin.