We construct a bounded degree graph G, such that a simple random walk on it is transient but the random walk path (i.e., the subgraph of all the edges the random walk has crossed) has only finitely many cutpoints, almost surely. We also prove that the expected number of cutpoints of any transient Markov chain is infinite. This answers two questions of James, Lyons and Peres [4] . Additionally, we consider a simple random walk on a finite connected graph G that starts at some fixed vertex x and is stopped when it first visits some other fixed vertex y. We provide a lower bound on the expected effective resistance between x and y in the path of the walk, giving a partial answer to a question raised in [2] .
Benjamini, Gurel-Gurevich and Schramm REF constructed a bounded degree graph satisfying that the trace of SRW on it a.s.
Cutpoints and resistance of random walk paths
Annals of Probability 2011, Vol. 39, No. 3, 1122-1136
Mathematics