Perturbed Divisible Sandpiles and Quadrature Surfaces

Abstract

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 Z d (d ≥ 2) which continuously deforms occupied regions of the divisible sandpile model of Levine and Peres (J. Anal. Math. 111(1), 151-219 2010), by redistributing the total mass of the system onto 1 msub-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 singular perturbation, which is known to converge to a Bernoulli type free boundary. We prove that models, generated from a single source, have a scaling limit, if the threshold m is fixed. Moreover, this limit is a ball, and the entire mass of the system is being redistributed onto an annular ring of thickness 1 m . By compactness argument we show that when m tends to infinity sufficiently slowly with respect to the scale of the model, then in this case also there is scaling limit which is a ball, with the mass of the system being uniformly distributed onto the boundary of that ball, and hence we recover a quadrature surface in this case. Depending on the speed of decay of 1/m, the visited set of the sandpile interpolates between spherical and polygonal shapes. Finding a precise characterisation of this shape-transition phenomenon seems to be a considerable challenge, which we cannot address at this moment.

Discrete Balayage and Boundary Sandpile

“…We have, due to Harnack's inequality, that u n (x (1) ) ≥ Cu n (x (0) ) with a constant C = C(d, r 0 ), hence it is enough to prove the lemma for x (0) replaced by x (1) . For that, we apply the same argument as we had for x (0) to x (1) .…”

“…We have, due to Harnack's inequality, that u n (x (1) ) ≥ Cu n (x (0) ) with a constant C = C(d, r 0 ), hence it is enough to prove the lemma for x (0) replaced by x (1) . For that, we apply the same argument as we had for x (0) to x (1) . Since x (1) has at most d − 1 non-zero coordinates, this reduction procedure on coordinates will terminate in at most d − 1 number of steps, where in the last step, the corresponding cylinder (4.16) will intersect the ball Z α 0 n 1/d implying the desired estimate of the lemma.…”

“…(iv) if we set u 0 (0) = +∞, and let the set of vectors N be defined as in (2.9), then for any x (1) , x (2) ∈ R d with the property that x (2) − x (1) is non-zero and is collinear to any of the vectors of N, we have…”

“…Next, relying on monotonicity of the odometer, one can show that the visited set of the ASM contains a diamond (a convex hull of the points {±e i , i = 1, ..., d} restricted to Z d ) scaled by a factor of R ≈ n 1/d , and is contained in a cube of size R ≈ n 1/d (cf. [1,Proposition 3.7]). Of course this result is not new, and moreover the constants we can get in these embeddings are slightly worse than the ones obtained by Fey and Redig [12], and imporved later by Levine and Peres [24].…”

“…In many other (related) free boundary value problems (known as Bernoulli type) the zero boundary gradient |∇u| = 0 in the above takes a different turn, and is a prescribed (strictly) non-zero function, and the volume potential U χ D in (1.2) is replaced by surface/single layer potential (of the a priori unknown domain) U χ ∂D . In terms of sandpile redistribution this means to find a domain D such that the 1 The term partial-balayage was first coined by Ognyan Kounchev as was pointed to us by Björn Gustafsson at KTH. 2 Actually Zidarov (at page 109 in his book) claims to prove that the model is abelian.…”