Given a discrete random walk on a finite graph G, the vacant set and vacant
net are, respectively, the sets of vertices and edges which remain unvisited by
the walk at a given step t.%These sets induce subgraphs of the underlying
graph. Let Ξ(t) be the subgraph of G induced by the vacant set of the
walk at step t. Similarly, let Ξ(t) be the subgraph of G
induced by the edges of the vacant net. For random r-regular graphs Grβ, it
was previously established that for a simple random walk, the graph Ξ(t)
of the vacant set undergoes a phase transition in the sense of the phase
transition on Erd\H{os}-Renyi graphs Gn,pβ. Thus, for rβ₯3 there is an
explicit value tβ=tβ(r) of the walk, such that for tβ€(1βΟ΅)tβ,
Ξ(t) has a unique giant component, plus components of size O(logn),
whereas for tβ₯(1+Ο΅)tβ all the components of Ξ(t) are of
size O(logn). We establish the threshold value t for a phase
transition in the graph Ξ(t) of the vacant net of a simple
random walk on a random r-regular graph. We obtain the corresponding
threshold results for the vacant set and vacant net of two modified random
walks. These are a non-backtracking random walk, and, for r even, a random
walk which chooses unvisited edges whenever available. This allows a direct
comparison of thresholds between simple and modified walks on random
r-regular graphs. The main findings are the following: As r increases the
threshold for the vacant set converges to nlogr in all three walks. For
the vacant net, the threshold converges to rn/2logn for both the simple
random walk and non-backtracking random walk. When rβ₯4 is even, the
threshold for the vacant net of the unvisited edge process converges to rn/2,
which is also the vertex cover time of the process.Comment: Added results pertaining to modified walk