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 ASM with a modified initial configuration. We also prove that 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.

中文翻译：

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泄漏阿贝尔沙堆模型的极限形状

泄漏阿贝尔沙堆模型 (Leaky-ASM) 是一种增长模型，其中 $n$ 个沙粒从 $\mathbb{Z}^2$ 的原点开始，并根据倾倒规则沿顶点扩散。如果一个站点的沙子量超过阈值，它可能会倒塌。在每次倒塌中，一个站点向每个邻居发送一些沙子，并泄漏一部分 1-1 美元/天的沙子。在对称情况下，我们将极限形状计算为 $d$ 的函数，其中每个倒塌向每个邻居发送等量的沙子。极限形状收敛为$d\to 1$ 的圆和$d\to \infty $ 的菱形。我们通过比较一个站点的里程表函数与一个被杀死的随机游走在该站点死亡的概率来计算极限形状。当$d\to 1$ 时，Leaky-ASM 收敛到具有修改的初始配置的 ASM。我们还证明了极限形状是一个圆，当与 $n\to \infty $ 同时我们有 $d=d_n$ 收敛到 $1$ 比任何 $n$ 的幂慢。要获得有关 ASM 的信息，需要更快的收敛速度。