Skip to main content

On the Widom–Rowlinson Occupancy Fraction in Regular Graphs


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.

Hide All
[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.
[2] Chayes, J., Chayes, L. and Kotecky, R. (1995) The analysis of the Widom–Rowlinson model by stochastic geometric methods. Comm. Math. Phys. 172 551569.
[3] Davies, E., Jenssen, M., Perkins, W. and Roberts, B. (2015) Independent sets, matchings, and occupancy fractions. arXiv:1508.04675
[4] Dembo, A., Montanari, A., Sun, N., et al. (2013) Factor models on locally tree-like graphs. Ann. Probab. 41 41624213.
[5] Galvin, D. (2013) Maximizing H-colorings of a regular graph. J. Graph Theory 73 6684.
[6] Galvin, D. (2014) Three tutorial lectures on entropy and counting. arXiv:1406.7872
[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.
[8] Kahn, J. (2001) An entropy approach to the hard-core model on bipartite graphs. Combin. Probab. Comput. 10 219237.
[9] Lebowitz, J. and Gallavotti, G. (1971) Phase transitions in binary lattice gases. J. Math. Phys. 12 11291133.
[10] Radhakrishnan, J. (2003) Entropy and counting. In Computational Mathematics, Modelling and Algorithms (Mishra, J. C., ed), Vol. 146, Narosa Publishers.
[11] Ruelle, D. (1971) Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27 1040.
[12] Sernau, L. (2015) Graph operations and upper bounds on graph homomorphism counts. arXiv:1510.01833
[13] Widom, B. and Rowlinson, J. S. (1970) New model for the study of liquid–vapor phase transitions. J. Chem. Phys. 52 16701684.
[14] Zhao, Y. (2010) The number of independent sets in a regular graph. Combin. Probab. Comput. 19 315320.
[15] Zhao, Y. (2011) The bipartite swapping trick on graph homomorphisms. SIAM J. Discrete Math. 25 660680.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 47 *
Loading metrics...

Abstract views

Total abstract views: 259 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd June 2018. This data will be updated every 24 hours.