We study systems of repelling particles living on the boundary of a Galton–Watson branching process. The profinite completion of such a tree carries a natural ultrametric determined by the branching structure, and the interaction energy of a pair of particles is taken to be minus the logarithm of this ultrametric distance. This extends the familiar logarithmic interaction on regular non-archimedean trees to a random hierarchical environment. Because the underlying tree is random, the canonical and grand canonical partition functions are random as well. We consider their expectations over the law of the branching process and show that these mean partition functions satisfy recursive identities that reflect the hierarchical factorization of the tree. For the canonical ensemble we derive a recursion in the particle number which implies that, when the offspring distribution takes only finitely many values q, the mean partition functions are rational functions of the corresponding quantities q−β. For the grand canonical ensemble we obtain a functional equation expressing its generating function in terms of the offspring distribution. These results provide a solvable model of logarithmically interacting particles on random ultrametric spaces and suggest natural extensions of non-archimedean Coulomb gases and their associated Igusa zeta-type structures.
C. D. Sinclair (Sun,) studied this question.