The random-cluster model on a homogeneous tree
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 \(\mathbb{Z}^d\).
scientific article; zbMATH DE number 849095