# Perturbed Divisible Sandpiles and Quadrature Surfaces

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## 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 \(\mathbb {Z}^{d}\) (*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 \(\frac 1m\)-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 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 \(\frac 1m\). 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.

## Keywords

Singular perturbation Lattice growth model Quadrature surface Bernoulli free boundary Boundary sandpile Balayage Divisible sandpile Scaling limit## Mathematics Subject Classification (2010)

31C20 35B25 35R35 (31C05 82C41)## Notes

### Acknowledgments

H. A. was supported by postdoctoral fellowship from Knut and Alice Wallenberg Foundation. H. Sh. was partially supported by Swedish Research Council.

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