# Perturbed Divisible Sandpiles and Quadrature Surfaces

- 40 Downloads

## 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.

## References

- 1.Aleksanyan, H., Shahgholian, H.: Discrete balayage and boundary sandpile. Journal d’Analyse Mathématique (to appear) preprint arXiv:1607.01525
- 2.Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of the 1/
*f*noise. Phys. Rev. Lett.**59.4**, 381 (1987)CrossRefGoogle Scholar - 3.Berestycki, H., Caffarelli, L.A., Nirenberg, L.: Uniform Estimates for Regularization of Free Boundary Problems. Analysis and Partial Differential Equations, 567–619, Lecture Notes in Pure and Appl Math., p 122. Dekker, New York (1990)Google Scholar
- 4.Fukai, Y., Uchiyama, K.: Potential kernel for two-dimensional random walk. Ann. Probab.
**24**(4), 1979–1992 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Gustafsson, B., Shapiro, H.S.: What is a Quadrature Domain? Quadrature Domains and their Applications, 1-25, Oper. Theory Adv Appl., p 156. Basel, Birkhäuser (2005)Google Scholar
- 6.Heinonen, J.: Lectures on Lipschitz Analysis. University of Jyväskylä (2005)Google Scholar
- 7.Henrot, A.: Subsolutions and supersolutions in a free boundary problem. Ark. Mat.
**32**(1), 79–98 (1994)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Lawler, G.: Intersections of Random Walks, (Probability and Its Applications). Birkhäuser. Reprint of the 1996 Edition (2013)Google Scholar
- 9.Levine, L., Peres, Y.: Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile. Potential Anal.
**30**(1), 1–27 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Levine, L., Peres, Y.: Scaling limits for internal aggregation models with multiple sources. J. Anal. Math.
**111**(1), 151–219 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Lionel, L., Peres, Y.: Laplacian growth, sandpiles, and scaling limits. Bull. AMS (to appear), preprint at arXiv:1611.00411 (2016)
- 12.Levine, L.: Limit Theorems for Internal Aggregation Models, PhD thesis University of California Berkley (2007)Google Scholar
- 13.Pegden, W., Smart, C.K.: Convergence of the Abelian sandpile. Duke Math. J.
**162**(4), 627–642 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Petrosyan, A., Shahgholian, H., Uraltseva, N.: Regularity of free boundaries in obstacle-type problems AMS (2012)Google Scholar
- 15.Shahgholian, H.: Existence of quadrature surfaces for positive measures with finite support. Potential Anal.
**3**(2), 245–255 (1994)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Shahgholian, H.: Quadrature surfaces as free boundaries. Ark. Mat.
**32**(2), 475–492 (1994)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Uchiyama, K.: Green’s functions for random walks on \(\mathbb {Z}^{N}\). Proc. Lond. Math. Soc.
**77**(1), 215–240 (1998)MathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.