Source: J. Appl. Probab. Volume 42, Number 1
(2005), 295-301.
We consider a random walker on a d-regular graph. Starting
from a fixed vertex, the first step is a unit step in any one of
the d directions, with common probability 1/d for
each one. At any later step, the random walker moves in any one of
the directions, with probability q for a reversal of
direction and probability p for any other direction. This
model was introduced and first studied by Gillis (1955), in the
case when the graph is a d-dimensional square lattice. We
prove that the Gillis random walk on a d-regular graph is
recurrent if and only if the simple random walk on the graph is
recurrent. The Green function of the Gillis random walk will be
also given, in terms of that of the simple random walk.
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