# Laplacian growth, sandpiles, and scaling limits

@article{Levine2016LaplacianGS, title={Laplacian growth, sandpiles, and scaling limits}, author={Lionel Levine and Yuval Peres}, journal={Bulletin of the American Mathematical Society}, year={2016}, volume={54}, pages={355-382} }

Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress over the last decade on discrete models of (internal) Laplacian growth, including the abelian sandpile, internal DLA, rotor aggregation, and the scaling limits of these models on the lattice Z^d as the mesh size goes to zero. These models provide a window into the tools of discrete potential theory…

## 18 Citations

Laplacian growth and sandpiles on the Sierpiński gasket: limit shape universality and exact solutions

- Physics, Mathematics
- 2018

We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner…

My title

- 2021

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…

Tropical curves in sandpile models

- Mathematics
- 2015

A sandpile is a cellular automata on a subgraph $\Omega_h$ of ${h}\mathbb Z^2$ which evolves by the toppling rule: if the number of grains at a vertex is at least four, then it sends one grain to…

DIVISIBLE SANDPILE ON SIERPINSKI GASKET GRAPHS

- MathematicsFractals
- 2019

The divisible sandpile model is a growth model on graphs that was introduced by Levine and Peres [Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, Potential Anal.…

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

- Mathematics, Physics
- 2019

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…

Averaging Principle and Shape Theorem for a Growth Model with Memory

- Mathematics
- 2018

We present a general approach to study a class of random growth models in $n$-dimensional Euclidean space. These models are designed to capture basic growth features which are expected to manifest at…

Non-Archimedean Coulomb gases

- Physics, Mathematics
- 2019

This article aims to study the Coulomb gas model over the $d$-dimensional $p$-adic space. We establish the existence of equilibria measures and the $\Gamma$-limit for the Coulomb energy functional…

Sandpile toppling on Penrose tilings: identity and isotropic dynamics

- Physics
- 2020

We present experiments of sandpiles on grids (square, triangular, hexagonal) and Penrose tilings. The challenging part is to program such simulator; and our javacript code is available online, ready…

Perfect boundaries in rotor-router aggregation on cylinders

- Mathematics, Physics
- 2016

We study a rotor-router version of the internal diffusion-limited aggregation introduced by J.Propp. The existing estimations of boundary fluctuations of the aggregation cluster show that they grow…

A shape theorem for exploding sandpiles

- Mathematics
- 2021

We study scaling limits of exploding Abelian sandpiles using ideas from percolation and front propagation in random media. We establish sufficient conditions under which a limit shape exists and show…

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