Random walk on the range of random walk

We study the random walk $X$ on the range of a simple random walk on $\mathbb{Z}^d$ in dimensions $d\geq 4$. When $d\geq 5$ we establish quenched and annealed scaling limits for the process $X$, which show that the intersections of the original simple random walk path are essentially unimportant. For $d=4$ our results are less precise, but we are able to show that any scaling limit for $X$ will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when $d=4$ similar logarithmic corrections are necessary in describing the asymptotic behaviour of the return probability of $X$ to the origin.