论文标题
泄漏的Abelian沙珀模型的极限形状
The Limit Shape of the Leaky Abelian Sandpile Model
论文作者
论文摘要
泄漏的Abelian Sandpile模型（漏水）是一种增长模型，其中$ n $ sand粒的砂粒子以$ \ m athbb {z}^2 $开始，并根据倒塌规则沿顶点延伸。如果地点的沙子量高于阈值，则可以推翻。在每个座位中，一个站点向每个邻居发送一些沙子，并泄漏一部分$ 1-1/d的沙子。在对称情况下，我们计算限制形状作为$ d $的函数，在对称情况下，每个篮板都会向每个邻居发送同等数量的沙子。极限形状将圆圈收敛为$ d \至1 $，而钻石则为$ d \ to \ infty $。我们通过比较位点的里程表函数与杀死的随机行走在该站点死亡的概率来计算极限形状。当$ d \至1 $时，泄漏 - asm会通过修改的初始配置收敛到Abelian Sandpile模型（ASM）。我们还证明，当与$ n \ to \ infty $同时，我们的限制形状是一个圆形，我们有$ d = d_n $收敛到$ 1 $比$ n $的任何功率慢1 $。为了获取有关ASM更快收敛的信息。
The leaky abelian sandpile model (Leaky-ASM) is a growth model in which $n$ grains of sand start at the origin in $\mathbb{Z}^2$ and diffuse along the vertices according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion $1-1/d$ of its sand. We compute the limit shape as a function of $d$ in the symmetric case where each topple sends an equal amount of sand to each neighbor. The limit shape converges to a circle as $d\to 1$ and a diamond as $d\to\infty$. We compute the limit shape by comparing the odometer function at a site to the probability that a killed random walk dies at that site. When $d\to 1$ the Leaky-ASM converges to the abelian sandpile model (ASM) with a modified initial configuration. We also prove the limit shape is a circle when simultaneously with $n\to\infty$ we have that $d=d_n$ converges to $1$ slower than any power of $n$. To gain information about the ASM faster convergence is necessary.
