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 HWR, a path on three vertices with a loop on each vertex, is maximised by K d+1 . This proves a conjecture of Galvin. The Widom-Rowlinson ModelA Widom-Rowlinson assignment or configuration on a graph G is a map χ : V (G) → {0, 1, 2} so that 1 and 2 are not assigned to neighbouring vertices, or in other words, a graph homomorphism from G to the graph H WR consisting of a path on 3 vertices with a loop on each vertex (the middle vertex represents the label 0). Call the set of all such assignments Ω(G). The Widom-Rowlinson model on G is a probability distribution over Ω(G) parameterised by λ ∈ (0, ∞), given by:where X i (χ) is the number of vertices coloured i under χ, andis the partition function. Evaluating P G (λ) at λ = 1 counts the number of homomorphisms from G to H WR . We think of vertices assigned 1 and 2 as "coloured" and those assigned 0 as "uncoloured" (see Figure 1).The Widom-Rowlinson model was introduced by Widom and Rowlinson in 1970 [13], as a model of two types of interacting particles with a hard-core exclusion between particles of different types: colour 1 and 2 represent particles of each type and colour 0 represents an unoccupied site. The model has been studied both on lattices [9] and in the continuum [11,2] and is known to exhibit a phase transition in both cases.The Widom-Rowlinson model is one case of a general random model: that of choosing a random homomorphism from a large graph G to a fixed graph H. In the Widom-Rowlinson case, we take H = H WR . Another notable case is H ind , an edge between two vertices,

Abstract. Let H WR be the path on 3 vertices with a loop at each vertex. D. Galvin [4,5] conjectured, and E. Cohen, W. Perkins and P. Tetali [2] proved that for any d-regular simple graph G on n vertices we haveIn this paper we give a short proof of this theorem together with the proof of a conjecture of Cohen, Perkins and Tetali [2]. Our main tool is a simple bijection between the Widom-Rowlinson model and the hard-core model on another graph. We also give a large class of graphs H for which we haveIn particular, we show that the above inequality holds if H is a path or a cycle of even length at least 6 with loops at every vertex.

We formulate and prove inverse mixing lemmas in the settings of simplicial complexes and k-uniform hypergraphs. In the hypergraph setting, we extend results of Bilu and Linial for graphs. In the simplicial complex setting, our results answer a question of Parzanchevski et al.

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