Title: Cut-off phenomenon for the abelian sandpile model on tiling graphs
Speaker: Robert Hough (Stony Brook University)
Abstract: In the abelian sandpile model on a graph $G = (V,E)$ with sink $s$, there are a non-negative number of chips $\sigma(v)$ at each non-sink vertex $v$. If $\sigma(v) \geq \deg(v)$, the vertex can 'topple' passing one chip to each neighbor. Any chips which fall on the sink are lost from the model. A configuration $\sigma$ is called 'stable' if no vertex can topple. In driven dynamics in the model, at each step a chip is added to the model at a uniform random vertex, and all legal topplings are performed until a stable configuration is reached. Together with Dan Jerison and Lionel Levine, I determined the asymptotic mixing time to stationarity and proved a cut-off phenomenon for dynamics on a square $N\times N$ grid with periodic boundary conditions and a single sink in the limit $N \to \infty$. Recently, with Hyojeong Son, I have extended this result to prove a cut-off phenomenon for sandpile dynamics on a growing piece of an arbitrary plane or space tiling, with open or periodic boundary condition, and proved that the asymptotic mixing time is equal in two dimensions subject to a reflection condition. A different boundary behavior exists for the D4 lattice in dimension 4, in which the open boundary can change the mixing time. I will discuss the spectral methods behind these results.