Laplacian growth, sandpiles, and scaling limits

Abstract

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: harmonic functions, martingales, obstacle problems, quadrature domains, Green functions, smoothing. We also present one new result: rotor aggregation in Z d has O(log r) fluctuations around a Euclidean ball, improving a previous power-law bound. We highlight several open questions, including whether these fluctuations are O(1).

How Hard is it to Predict Sandpiles on Lattices? A Survey

“…• allows to consider constant time basic operations, • preserves the "dynamical complexity", within P and with P-hard problems. Unbounded values bring considerations of another kind (namely, of computing fixed points from a single column of sand grains, as in [45,46,47,58]) not necessary to capture the intrinsic complexity of the problem. Cell positions are also admitted to be given using O(log(n d )) bits which is o(n d ) (see Lemma 3 for a polynomial bound on the most distant cell that can receive a grain).…”

Asymptotic Height Distribution in High-Dimensional Sandpiles

“…Sandpiles are a lattice model of self-organized criticality, introduced by Bak, Tang and Wiesenfeld [3], and have been studied in both physics and mathematics. See the surveys [10], [15], [23], [9], [6]. Although the model can easily be defined on an arbitrary finite connected graph, in this paper we will restrict to subsets of Z d .…”

Divisible Sandpile on Sierpinski Gasket Graphs

“…The limit shape of the divisible sandpile cluster was identified on ℤ in [LP09], on homogeneous trees in [Lev09], and on the comb lattice in [HS12]. See also the recent survey [LP16] for an introduction to the divisible sandpile model. The aim of this paper is to identify the limit shape of the divisible sandpile cluster on the doubly-infinite Sierpinski gasket graph SG, by making strong use of the property of SG of being finitely ramified, which means that it can be disconnected by removing a finite number of points.…”