Heat kernel for random walk trace on ℤ3 and ℤ4
We study the simple random walk on the range of simple random walk on ℤ3 and ℤ4. In dimension four, we establish quenched bounds for the heat kernel of and max0≤≤| | which require extra logarithmic correction terms to the higher-dimensional case. In dimension three, we demonstrate anomalous behavior of at the quenched level. In order to establish these estimates, we obtain several asymptotic estimates for cut times of simple random walk and asymptotic estimates for loop-erased...