Cut-off for sandpiles on tiling graphs

Exact sampling and fast mixing of Activated Random Walk

“…In contrast to (1), the abelian sandpile has an interval of critical means [9], and the problem of whether a sandpile on Z d stabilizes almost surely is not even known to be decidable [19]. The root cause of this nonuniversality is slow mixing: For example, the sandpile mixing time on both the ball B(0, n) ∩ Z d and on the torus Z d n is of order n d log n [11,12]. This extra log factor is responsible for the non-universality of the sandpile threshold state [18].…”

The spectrum of the abelian sandpile model

“…The article [19] provides an accessible introduction, and computes several sandpile statistics for varying graph geometries. In [14] and [13] the authors evaluated the spectral gap, asymptotic mixing time and proved a cutoff phenomenon in sandpile dynamics on a growing piece of a plane or space tiling given periodic or open boundary conditions. In particular, in the article [13] spectral factors related to the harmonic modulo 1 functions on the tiling are identified, and these factors are demonstrated to control the spectral gap and asymptotic mixing time of the dynamics.…”

“…In [14] and [13] the authors evaluated the spectral gap, asymptotic mixing time and proved a cutoff phenomenon in sandpile dynamics on a growing piece of a plane or space tiling given periodic or open boundary conditions. In particular, in the article [13] spectral factors related to the harmonic modulo 1 functions on the tiling are identified, and these factors are demonstrated to control the spectral gap and asymptotic mixing time of the dynamics. The purpose of this article is to describe a general method of calculating the spectral factors numerically, and to perform this calculation in specific examples.…”

“…Let C 1 pT q denote the set of integer valued functions on T which are finitely supported and have sum 0. In [13] the spectral parameter of a tiling is defined to be (1) γ " inf # ÿ xPT 1 ´cosp2πξ x q : ∆ξ P C 1 pT q, ξ ı 0 mod 1…”

“…and Γ " max j Γ j . In [13], the following theorem is proved determining the asymptotic mixing time in terms of the spectral factor. Write T m " T {mΛ for the periodic tiling graph and let T m be the graph formed by giving a fundamental domain for T {tH i,mj , 1 ď i ď d, j P Zu an open boundary condition.…”