The dependency digraph for irreducible finite-range random walk in free groups

This article is one of a triptych composed with Che25a and Che25b, that aims at proving an asymptotic expansion to any order of the passage probability of an irreducible equivariant finite range random walk on a tree. In this text we study the analytic and geometric properties of the objects introduced in Che25b. To do so we introduce the definition of a ''flooded cavern tree'' enabling a precise study of some dependency digraph associated with a given irreducible finite-range random walk on a free group.