  We establish scaling limits for the random walk whose state space is the
range of a simple random walk on the four-dimensional integer lattice. These
concern the asymptotic behaviour of the graph distance from the origin and the
spatial location of the random walk in question. The limiting processes are the
analogues of those for higher-dimensional versions of the model, but additional
logarithmic terms in the scaling factors are needed to see these. The proof
applies recently developed machinery relating the scaling of resistance metric
spaces and stochastic processes, with key inputs being natural scaling
statements for the random walk's invariant measure, the associated effective
resistance metric, the graph distance, and the cut times for the underlying
simple random walk.
