### Electrical resistance of the trace of branching random walk

Probability Seminar

6th May 2022, 3:30 pm – 4:30 pm

Fry Building, 2.04

Motivated by the problem of simple random walk on certain recurrent random fractal

graphs, such as critical percolation clusters, we consider the simplified setting where

the graph is given by the space-time trace of a critical branching random walk (BRW)

conditioned to survive forever. One possible approach to studying the random walk on

such a graph is via bounds on the volume growth and electrical resistance of the graph,

of which the resistance estimates are the main difficulty in the case of the BRW trace.

After a brief overview of related results, we present the main ideas of J & Nachmias (2014),

who showed that the resistance in dimensions d<6 differs from the mean-field scaling
by at least a power. In parallel, we discuss the results of J & Lopez (2020) who extended the
above to show that in the critical dimension d=6 there is at least a logarithmic correction
to the mean-field scaling. Finally, we turn to the prospects of obtaining lower bounds
on the resistance, that is largely open below the critical dimension, as well as other
open problems.

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