We establish a formula for the distribution of loop-erased random walks at certain random times. Several classical results on spanning trees, including Wilson's algorithm, follow easily, as well as a method to construct random Hamiltonian cycles.
"Loop-Erased Random Walks, Spanning Trees and Hamiltonian Cycles." Electron. Commun. Probab. 5 39 - 50, 2000. https://doi.org/10.1214/ECP.v5-1016