My research interests lie at the intersection of probability and combinatorics, with some motivations from statistical physics. I have done research on various stochastic processes that exhibit self-organized criticality, building a unifying theory for those processes. I have also worked on random walks in random environments that interact with the walker, where the uniform spanning forest makes a surprise appearance.
We construct, for any graph, a rotor configuration for which its escape rate is equal to the escape rate of simple random walk, and thus answer a question of Florescu, Ganguly, Levine, and Peres (2014).
We study rotor walks on transient graphs with initial rotor configuration sampled from the oriented wired uniform spanning forest (OWUSF) measure.
Among other things, we give a simple sufficient and necessary condition for the OWUSF measure to be a stationary distribution for the rotor walk.
We construct a bijection between (1) the necklaces of length n with 2 colors, and (2) the sets of integers modulo n with subset sum divisible by n, provided that n is odd, and thus answers a bijective problem posed by Richard Stanley (Enumerative Combinatorics Vol. 1 Chapter 1, Problem 105(b)).
We show that, for a strongly-connected digraph, the generating function of recurrent configurations of the sandpile model by the number of chips is
an invariant that does not depend on the choice of the sink of the sandpile model, and thus answers a conjecture of Perrot and Pham (2013).
We study the multi-tiling problem by a convex polytope, where the tiling set is a finite union of translated lattices.
Under a mild condition, we show that the tiling set can be replaced with a lattice, and is a step in the direction of proving the conjecture of Gravin, Robins, and Shiryaev (2012).
We compute the sandpile group of generalized de Bruijn graphs and generalized Kautz graphs, and in the former case we relate this group to a quotient of the group of circulant matrices over a finite field. This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS.