爆炸沙堆的形状定理

Perturbed divisible sandpiles and quadrature surfaces

The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a new model of Laplacian growth on the lattice $\mathbb{Z}^d$ $(d\geq 2)$ which continuously deforms occupied regions of the \emph{divisible sandpile} model of Levine and Peres, by redistributing the total mass of the system onto $\frac 1m$-sub-level sets of the odometer which is a function counting total emissions of mass from lattice vertices. In free boundary terminology this goes in parallel with s…

单源随机沙堆模型中$p$-翻转的极限形状

Limit shape of single-source stochastic sandpiles with $p$-topplings on $\mathbb{Z}$

研究单源随机沙堆模型中$p$-翻转的极限形状，证明当粒子数趋于无穷时，最终稳定配置的宏观极限形状对称区间围绕原点，并通过中心极限定理证明边界波动收敛于高斯分布。

多层渗漏阿贝尔沙堆模型的极限形状

Limit shape of the leaky Abelian sandpile model with multiple layers

本文在任意维度上推广渗漏阿贝尔沙堆模型，引入多层颜色与任意倾倒分布，证明极限形状存在，并展示其随参数变化收敛到多面体或椭球，且具有二元性。

Odometers of Divisible Sandpile Models: Scaling Limits, iDLA and Obstacle Problems. A Survey

The divisible sandpile model is a fixed-energy continuous counterpart of the Abelian sandpile model. We start with a random initial configuration and redistribute mass deterministically. Under certain conditions the sandpile will stabilize. The associated odometer function describes the amount of mass emitted from each vertex during stabilization. In this survey we describe recent scaling limit results of the odometer function depending on different initial configurations and redistribution rules. Moreover we review connections to the obstacle problem from potential theory, including the conne…

Limiting shapes for deterministic centrally seeded growth models

We study the rotor router model and two deterministic sandpile models. For the rotor router model in $\mathbb{Z}^d$, Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter $h$ (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension $d \geq 1$. For the rotor router model, the limiting shape is a sphere for all values of $h$. For one of the sandpile models, and $h=2d-2$ (the maximal val…

Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work. For the shape consisting of $n=ω_d r^d$ sites, where $ω_d$ is the volume of the unit ball in $\R^d$, we show that the inradius of the set of occupied sites is at least $r-O(\log r)$, while the outradius is at most $r+O(r^α)$ for any $α> 1-1/d$. For a related model, the divisible sandpile, we show that the domain of occupied site…

Limit Theorems for Internal Aggregation Models

We study the scaling limits of three different aggregation models on the integer lattice Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a determini…

mated-CRT 地图上的内部 DLA

Internal DLA on mated-CRT maps

本文研究 mated-CRT 随机平面图上的内部 DLA，证明其形状收敛到 LQG 调和球，并为可分沙堆建立类似结论。