Probability Theory and Related Fields

, Volume 104, Issue 2, pp 231–253 | Cite as

The random-cluster model on a homogeneous tree

  • Olle Häggström
Article

Summary

The random-cluster model on a homogeneous tree is defined and studied. It is shown that for 1≦q≦2, the percolation probability in the maximal random-cluster measure is continuous inp, while forq>2 it has a discontinuity at the critical valuep=pc(q). It is also shown that forq>2, there is nonuniqueness of random-cluster measures for an entire interval of values ofp. The latter result is in sharp contrast to what happens on the integer lattice Zd.

Mathematics Subject Classification (1991)

60K35 82B20 05C80 

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Olle Häggström
    • 1
  1. 1.Department of MathematicsChalmers University of TechnologyGöteborgSweden

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