The Divisible Sandpile at Critical Density |
| |
Authors: | Lionel Levine Mathav Murugan Yuval Peres Baris Evren Ugurcan |
| |
Institution: | 1.Department of Mathematics,Cornell University,Ithaca,USA;2.Department of Mathematics,University of British Columbia and Pacific Institute for the Mathematical Sciences,Vancouver,Canada;3.Microsoft Research,Redmond,USA |
| |
Abstract: | The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses ≤ 1. The process stabilizes almost surely if m < 1 and it almost surely does not stabilize if m > 1, where m is the mean mass per vertex. The main result of this paper is that in the critical case m = 1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete bi-Laplacian Gaussian field. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|