2014/02/17 by A. D. Barbour, Barbour, A. D., Andrea Collevecchio +1
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1402.4163
openalex publication_date 2014/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the motion of a particle on a Galton Watson tree, when the probabilities of jumping from a vertex to any one of its neighbours is determined by a random process. Given the tree, positive weights are assigned to the edges in such a way that, viewed along any line of descent, they evolve as a random process. In order to introduce our method for proving transience or recurrence, we first suppose that the weights are i.i.d., reproving a result of Lyons and Pemantle. We then extend the argument to allow a Markovian environment, and finally to a random walk on a Markovian environment that changes the environment. Our approach involves studying the typical behaviour of processes on fixed lines of descent, which we then show determines the behaviour of the process on the whole tree.