Genealogy in critical generations of a diffusive random walk in random environment on trees

2024/10/10 by Alexis Kagan, Kagan, Alexis
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #advanced mathematical theories #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2410.08402

Abstract

We consider the range R⁽ⁿ⁾, the tree made up of visited vertices by a diffusive null-recurrent randomly biased walk \mathbbX on a Galton-Watson tree \mathbbT up to the n-th return time to its root and we consider the following genealogy problem: pick two vertices uniformly at random in a generation of order n in the tree R⁽ⁿ⁾. Where does the coalescence occur? it turns out that the coalescence happens either in the recent past or in the remote past.

Related

• Coalescence in small generations for the diffusive randomly biased walk on Galton-Watson trees
• Local times in critical generations of a random walk in random environment on trees
• The number of generations entirely visited for recurrent random walks on random environment
• Range and critical generations of a random walk on Galton-Watson trees
• Spread of visited sites of a random walk along the generations of a branching process