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On the Widom–Rowlinson Occupancy Fraction in Regular Graphs

Part of: Graph theory Equilibrium statistical mechanics

Published online by Cambridge University Press:  09 August 2016

EMMA COHEN ,
WILL PERKINS  and
PRASAD TETALI
EMMA COHEN
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: ecohen32@gatech.edu, tetali@math.gatech.edu)
WILL PERKINS
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK (e-mail: math@willperkins.org)
PRASAD TETALI
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: ecohen32@gatech.edu, tetali@math.gatech.edu)
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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, Kd+1. As a corollary we find that Kd+1 also maximizes the normalized partition function of the Widom–Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalized 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 maximized by Kd+1. This proves a conjecture of Galvin.

MSC classification

Primary: 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
Secondary: 05C35: Extremal problems 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 2 , March 2017 , pp. 183 - 194
Copyright
Copyright © Cambridge University Press 2016 

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