Electrical resistance of the trace of branching random walk
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.