On the Widom–Rowlinson Occupancy Fraction in Regular Graphs

Abstract

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,

Extremes of the internal energy of the Potts model on cubic graphs

“…In this paper, we derive tight bounds on U q G (β) for cubic (3-regular) graphs in the anti-ferromagnetic (β > 0) regime. From (1), these bounds immediately imply corresponding tight bounds on the free energy of the Ising and Potts models and hence the respective partition functions. We determine, for every q, the maximum and minimum of both the internal energy and the free energy per particle as well as the family of graphs that achieve these bounds.…”

“…This method builds on previous work on independent sets and matchings and the Widom‐Rowlinson model , but here we generalize the previous approach in two ways: (1) we deal with q ‐spin models instead of 2‐spin models; (2) we deal with soft and hard constraints instead of just hard constraints. The family of linear programs in for matchings was an infinite family of LP's indexed by two parameters — the vertex degree d and a fugacity parameter — and the entire family could be solved analytically with a single proof via LP duality.…”

Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

“…The first non-trivial case of Problem 2.5 where the maximizing G is not K d,d was obtained recently by Cohen, Perkins, and Tetali [11]. Theorem 2.8 (Cohen, Perkins, and Tetali [11]). For any d-regular graph G we have…”

“…Theorem 2.8 was initially proved [11] using the occupancy fraction method, which will be discussed in Section 7. Subsequently, a much shorter proof was given in [10] (also see Sernau [52]).…”

“…. , p d ) achieves the maximum of value of p 0 while satisfying the constraints (10), (11), and (12), then every inequality in (12) …”