梳状格上阿贝尔沙堆模型的极限形状、雪崩与稳定化
Limit shape, avalanches, and stabilisation of abelian sandpiles on comb lattices
AI总结：
本文研究梳状格上阿贝尔沙堆的平稳测度支撑、雪崩形状及单源极限形状，证明其与可除沙堆等模型的极限形状一致，确立了普适性。
AI中文摘要：
我们研究了梳状格上阿贝尔沙堆模型的两个方面。首先，我们证明了平稳测度的无限体积极限支撑在饱和构型上。然后，我们研究了在原点周围大小为$(2n+1)\times(2n+1)$的有限盒子中，从原点添加一个粒子所引发的雪崩形状。在这些有限盒子上的平稳分布中，我们证明了雪崩以趋于$1$的概率沿着垂直齿到达边界，而水平扩散的阶为$\sqrt{n}$。最后，我们确立了梳状格上阿贝尔沙堆的单源极限形状与可除沙堆、内部扩散限制聚集（IDLA）和转子路由器聚集模型的相应极限形状一致。这确立了梳状格上的极限形状普适性。
英文摘要：
We study two aspects of the abelian sandpile model on comb lattices. First, we prove that the infinite-volume limit of the stationary measures is supported on the saturated configuration. We then investigate the shape of avalanches induced by adding a particle at the origin in finite boxes of size $(2n+1)\times(2n+1)$ around the origin. In the stationary distribution on these finite boxes, we show that avalanches reach the boundary along the vertical teeth with probability tending to $1$, while the horizontal spread is of order $\sqrt{n}$. Finally, we establish that the single-source limit shape for abelian sandpiles on the comb lattice agrees with the corresponding limit shapes for the divisible sandpile, internal diffusion-limited aggregation (IDLA), and rotor-router aggregation models. This establishes limit shape universality on the comb lattice.
