Hostname: page-component-6bb9c88b65-kzqxb Total loading time: 0 Render date: 2025-07-24T04:00:55.379Z Has data issue: false hasContentIssue false
  • English
  • Français

Percolation of Hard Disks

Part of: Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  30 January 2018

D. Aristoff
D. Aristoff*
Affiliation:
University of Minnesota
*
∗ Postal address: Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, MN 55455, USA, Email address: daristof@umn.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Random arrangements of points in the plane, interacting only through a simple hard-core exclusion, are considered. An intensity parameter controls the average density of arrangements, in analogy with the Poisson point process. It is proved that, at high intensity, an infinite connected cluster of excluded volume appears almost surely.

Keywords

PercolationPoisson point processGibbs measuregrand canonical Gibbs distributionstatistical mechanicshard spherehard diskexcluded volumegas/liquid transitionphase transition

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B43: Percolation 82B26: Phase transitions (general)

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 235 - 246
Copyright
© Applied Probability Trust 

References

Bernard, E. P. and Krauth, W. (2011). Two-step melting in two dimensions: first-order liquid-hexatic transition. Phys. Rev. Lett. 107, 155704.Google Scholar
Bowen, L., Lyons, R., Radin, C. and Winkler, P. (2006). A solidification phenomenon in random packings. SIAM J. Math. Anal. 38, 1075–1089.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin.Google Scholar
Georgii, H.-O. (1995). The equivalence of ensembles for classical systems of particles. J. Statist. Phys. 80, 1341–1378.Google Scholar
Griffiths, R. B. (1964). Peierls proof of spontaneous magnetization in a two-dimensional Ising ferromagnet. Phys. Rev. 136, A437–A439.CrossRefGoogle Scholar
Kratky, K. W. (1988). Is the percolation transition of hard spheres a thermodynamic phase transition? J. Statist. Phys. 52, 1413–1421.Google Scholar
Lanford, O. E., III and Ruelle, D. (1969). Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys. 13, 194–215.Google Scholar
Lebowitz, J. L., Mazel, A. E. and Presutti, E. (1998). Rigorous proof of a liquid-vapor phase transition in a continuum particle system. Phys. Rev. Lett. 80, 4701–4704.Google Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.CrossRefGoogle Scholar
Richthammer, T. (2009). Translation invariance of two-dimensional Gibbsian systems of particles with internal degrees of freedom. Stoch. Process. Appl. 119, 700–736.Google Scholar
Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159.CrossRefGoogle Scholar
Woodcock, L. V. (2012). Percolation transitions in the hard-sphere fluid. AIChE J. 58, 1610–1618.Google Scholar