Hostname: page-component-89b8bd64d-x2lbr Total loading time: 0 Render date: 2026-05-14T06:04:18.000Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Widom–Rowlinson Occupancy Fraction in Regular Graphs

Part of: Equilibrium statistical mechanics Graph theory

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)
Get access Rights & Permissions [Opens in a new window]

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. As a corollary we find that K d+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 K d+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

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 2 , March 2017 , pp. 183 - 194
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Brightwell, G. and Winkler, P. (2002) Hard constraints and the Bethe lattice: Adventures at the interface of combinatorics and statistical physics. In Proc. International Congress of Mathematicians, Vol. III, pp. 605–624.Google Scholar
[2] Chayes, J., Chayes, L. and Kotecky, R. (1995) The analysis of the Widom–Rowlinson model by stochastic geometric methods. Comm. Math. Phys. 172 551–569.Google Scholar
[3] Davies, E., Jenssen, M., Perkins, W. and Roberts, B. (2015) Independent sets, matchings, and occupancy fractions. arXiv:1508.04675 Google Scholar
[4] Dembo, A., Montanari, A., Sun, N., et al. (2013) Factor models on locally tree-like graphs. Ann. Probab. 41 4162–4213.CrossRefGoogle Scholar
[5] Galvin, D. (2013) Maximizing H-colorings of a regular graph. J. Graph Theory 73 66–84.Google Scholar
[6] Galvin, D. (2014) Three tutorial lectures on entropy and counting. arXiv:1406.7872 Google Scholar
[7] Galvin, D. and Tetali, P. (2004) On weighted graph homomorphisms. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 63, pp. 97–104.Google Scholar
[8] Kahn, J. (2001) An entropy approach to the hard-core model on bipartite graphs. Combin. Probab. Comput. 10 219–237.CrossRefGoogle Scholar
[9] Lebowitz, J. and Gallavotti, G. (1971) Phase transitions in binary lattice gases. J. Math. Phys. 12 1129–1133.Google Scholar
[10] Radhakrishnan, J. (2003) Entropy and counting. In Computational Mathematics, Modelling and Algorithms (Mishra, J. C., ed), Vol. 146, Narosa Publishers.Google Scholar
[11] Ruelle, D. (1971) Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27 1040.Google Scholar
[12] Sernau, L. (2015) Graph operations and upper bounds on graph homomorphism counts. arXiv:1510.01833 Google Scholar
[13] Widom, B. and Rowlinson, J. S. (1970) New model for the study of liquid–vapor phase transitions. J. Chem. Phys. 52 1670–1684.Google Scholar
[14] Zhao, Y. (2010) The number of independent sets in a regular graph. Combin. Probab. Comput. 19 315–320.Google Scholar
[15] Zhao, Y. (2011) The bipartite swapping trick on graph homomorphisms. SIAM J. Discrete Math. 25 660–680.CrossRefGoogle Scholar