Loop soup percolation and boundary dimension on metric trees

Source abstract

We determine the percolation threshold of the Brownian loop soup on a transient locally finite weighted tree with proper cable metric and no killing. It equals $1-D_G/2$, where $D_G\in[0,2]$ is the Hausdorff dimension of the boundary in the metric induced by the normalized Green function. This gives an analogue of Lyons' branching-number formula for Bernoulli percolation, with Green distance replacing graph distance. Spherically symmetric trees with unit conductances and branching number two realize every threshold in $[1/2,1]$, while geometric conductances give explicit positive thresholds below one-half and a subdivided binary tree has threshold zero. Under the capacity assumption of Drewitz, Prévost and Rodriguez, we characterize the threshold one-half by a finite-energy condition, without a minimum-degree assumption. Non-percolation at one-half then follows from their theorem and Lupu's coupling. For supercritical Galton-Watson trees with unit conductances, conditioned on survival, the boundary dimension is almost surely one and the threshold is one-half, with non-percolation at criticality. No offspring moment assumption is required, and leaves and vertices with one child are allowed. The classification also applies intrinsically to proper real trees spanned by their infinite rays.