  We consider a random object that is associated with both random walks and
random media, specifically, the superposition of a configuration of subcritical
Bernoulli percolation on an infinite connected graph and the trace of the
simple random walk on the same graph. We investigate asymptotics for the number
of vertices of the enlargement of the trace of the walk until a fixed time,
when the time tends to infinity. This process is more highly self-interacting
than the range of random walk, which yields difficulties. We show a law of
large numbers on vertex-transitive transient graphs. We compare the process on
a vertex-transitive graph with the process on a finitely modified graph of the
original vertex-transitive graph and show their behaviors are similar. We show
that the process fluctuates almost surely on a certain non-vertex-transitive
graph. On the two-dimensional integer lattice, by investigating the size of the
boundary of the trace, we give an estimate for variances of the process
implying a law of large numbers. We give an example of a graph with unbounded
degrees on which the process behaves in a singular manner. As by-products, some
results for the range and the boundary, which will be of independent interest,
are obtained.
