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We prove that the critical value of the one-dimensional Stochastic Sandpile Model is less than one. This verifies a conjecture of Rolla and Sidoravicius., Comment: 44 pages

We show that for small enough sleep rate, the critical density of the symmetric Activated Random Walk model on $\mathbb Z^2$ is strictly less than one.

The Aldous--Broder algorithm provides a way of sampling a uniformly random spanning tree for finite connected graphs using simple random walk. Namely, start a simple random walk on a connected graph and stop at the cover time. The tree formed by all the first-entrance edges has the law of a uniform spanning tree. Here we show that the tree formed by all the last-exit edges also has the law of a uniform spanning tree. This answers a question of Tom Hayes and Cris Moore from 2010. The proof relies on a bijection that is related to the BEST theorem in graph theory. We also give other applications of our results, including new proofs of the reversibility of loop-erased random walk, of the Aldous--Broder algorithm itself, and of Wilson's algorithm., Comment: 14 pages, 1 figure

We study the winding behavior of random walks on two oriented square lattices. One common feature of these walks is that they are bound to revolve clockwise. We also obtain quantitative results of transience/recurrence for each walk., Comment: 28 pages