  We prove that a simple random walk on quasi-transitive graphs with the volume
growth being faster than any polynomial of degree 4 has a.s. infinitely many
cut times, and hence infinitely many cutpoints. This confirms a conjecture
raised by I. Benjamini, O. Gurel-Gurevich and O. Schramm [2011, Cutpoints and
resistance of random walk paths, {\it Ann. Probab.} {\bf 39(3)}, 1122-1136]
that PATH of simple random walk on any transient vertex-transitive graph has
a.s. infinitely many cutpoints in the corresponding case.
