Random walks on random graphs are associated with diffusion phenomena in disordered media. This talk focuses on the simple random walk on uniform spanning trees (UST) and loop-erased random walks (LERW). The first two results are a quantitative estimate of the number of collisions and heat kernel fluctuations for the simple random walk on the three-dimensional UST. Next, I will discuss the annealed off-diagonal heat kernel of the simple random walk on high-dimensional LERWs.