The random-cluster model on a homogeneous tree
type: Item
instance of: scholarly article
title: The random-cluster model on a homogeneous tree
zbMATH Open document ID: 0838.60086
DOI: 10.1007/BF01247839
author: Olle Häggström
publication date: 1996-05-27T00:00:00Z
review text: The random-cluster model on a homogeneous tree is defined and studied. It is shown that for \(1 \leq q \leq 2\), the percolation probability in the maximal random-cluster measure is continuous in \(p\), while for \(q > 2\) it has a discontinuity at the critical value \(p = p_c (q)\). It is also shown that for \(q > 2\), there is nonuniqueness of random-cluster measures for an entire interval of values of \(p\). The latter result is in sharp contrast to what happens on the integer lattice \(\mathbb{Z}^d\).
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 05C80
zbMATH DE Number: 849095
zbMATH Keywords: random-cluster model
zbMATH Keywords: percolation probability
zbMATH Keywords: maximal random-cluster measure
zbMATH Keywords: nonuniqueness
MaRDI profile type: Publication
cites work: Discontinuity of the magnetization in one-dimensional \(1/| x-y| ^ 2\) Ising and Potts models.
cites work: Markov chains indexed by trees
cites work: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice.
cites work: The random-cluster model on the complete graph
cites work: Density and uniqueness in percolation
cites work: Q4848552
cites work: The stochastic random-cluster process and the uniqueness of random-cluster measures
cites work: Random-cluster measures and uniform spanning trees
cites work: Remarks on the limiting Gibbs states on a (d+1)-tree
cites work: The Ising model and percolation on trees and tree-like graphs
cites work: Random walks and percolation on trees
cites work: Markov random fields on an infinite tree
cites work: Countable state space Markov random fields and Markov chains on trees
full work available at URL: https://doi.org/10.1007/bf01247839
OpenAlex ID: W2086852576
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 849095