Once-excited random walks on general trees

We study once-excited random walks on general trees, modeled by placing a single "cookie" at each vertex. Each cookie acts as a metaphorical reward that is consumed upon the first visit to the vertex where the cookie is placed. On that initial visit, the walk is in an excited state and behaves like a biased random walk. Once the cookie is consumed, the process reverts to a symmetric random walk on all subsequent visits. We consider a random environment in which the bias parameters are independent random variables. We prove that the process exhibits a sharp phase transition between transience and recurrence on general trees with polynomial growth, where the critical threshold is determined by the branching-ruin number of the tree.