Scaling limit of a biased random walk on a critical branching random walk (Stochastic Analysis on Large Scale Interacting Systems)

Abstract

Relying on powerful resistance techniques developed in [8], the recent paper `Invariance principles for random walks in random environment on trees' [4] investigates random walks in random environment on tree-like spaces and their scaling limits in a certain regime, that is when the potential of the random walk in random environment converges. We introduce and summarise a result from [4]. We choose to review the example of a novel scaling continuum limit of a biased random walk on large critical branching random walk. In this case the diffusion that is not on natural scale is identified as a Brownian motion on a continuum random fractal tree with its canonical metric replaced by a distorted resistance metric. This example allows the least technical presentation (compared to the others covered in the main article). Moreover, it is nevertheless of current interest, given its relation to critical percolation.