2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72875Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of random-cluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree $T$ of a branching process. What is the probability that every infinite path of $T$, beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree?ProbabilityMathematical Physics60K35, 60J80, 82B20text