The aim of this paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). Given a measure $ \mu $ in $\mathbb{R}^d$ and a domain $\Omega$ containing the support of $\mu$, we call $\partial \Omega$ a quadrature surface for $\mu $ if one has $ \int_{\partial \Omega } h(x) d\mathcal{H}^{d-1} (x) = \int h(x) d\mu (x) $ for all functions $ h $ harmonic on $ \overline{\Omega} $
[Sh1] and [Sh2] by H. Shahgholian).
For example, if $\mu$ is a Dirac mass concentrated at some point $x\in \mathbb{R}^d$, then a sphere with center at $x$ and of an appropriate radius would give a quadrature surface. There is no complete description of measures $\mu$ admitting a quadrature surface identity, but in a special case of a finite sum of Dirac masses, for instance, the existence is due to Shahgholian.

In this paper we show that one can obtain a quadrature surface as a scaling limit of a certain growth model on the lattice $\mathbb{Z}^d $, where $ d\geq 2 $. Our motivation comes from an already established connection between quadrature domains (these are domains admitting a similar integral identity as above, but with integration against the Lebesgue measure over the domain instead of the surface) and Laplacian growth models by Levine and Peres (see [LP1] and [LP2] ).
They introduce a model called divisible sandpile, and prove that for initial distribution of finitely many points its scaling limit is a quadrature surface ( a smash sum of Euclidean balls, precisely, a concept after Diaconis and Fulton)
A shift from a domain to a lower-dimensional object, a hypersurface in our case, seems to necessitate an entirely different approach.

To define our model, start with a distribution of mass $\mu_0 $ on $ \mathbb{Z}^d $, i.e. a bounded non-negative function of finite support, let $n = \sum_{x\in \mathbb{Z}^d} \mu_0(x) $ be its total mass, and fix a threshold $ m>0 $. One by one we pick a vertex $x \in \mathbb{Z}^d$ and if it carries mass larger than $m$, distribute the excess of mass from $m$ evenly among the $2d$ lattice neighbors of $x$. Otherwise, if the mass carried by $x$ is bounded by $m$ but the cumulative emissions of mass from $x$ prior to a given epoch of time exceed $\frac 1m n^{2/d}$ distribute the entire mass of $x$ evenly among its lattice neighbors. If none of the two conditions hold for $x$ we move on to another vertex. This mass redistribution procedure is called toppling of $x$.
We show that provided any vertex of $\mathbb{Z}^d$ is chosen infinitely many times, the mass redistribution process will reach a stable state after countably many topplings, and regardless of the order of the topplings the final configuration will be the same (Abelian property). Informally, one may think about the proposed model, as a deformation of a divisible sandpile model of Levine and Peres, where we force the mass to accumulate on a $\frac 1m $-sub-level sets of the odometer. This model is partially inspired by singular perturbation problems in PDEs which are known to converge to Bernoulli type free boundaries.

The key concept in our analysis is the odometer function $u:\mathbb{Z}^d \rightarrow \mathbb{R}_+$ where for each $x\in \mathbb{Z}^d$ the value $u(x)$ shows total emissions of mass from $x$ during the lifetime of the process. In terms of this function, the discrete PDE problem solved by the model reads
$$\Delta u(x) + \mu_0(x) \leq m \ \ \ \text{ everywhere on } \mathbb{Z}^d $$
and
$$\Delta u(x) = m \mathbb{I}_{ \{ 0 < u < \frac{1}{m} n^{2/d} \} } (x) - \mu_0(x) \ \ \ \text{ for all } x \in \{u > 0\},$$
where $\Delta$ is the discrete (normalized) Laplacian. We show that the odometer $u$ is precisely the smallest super-solution to the above problem.
This characterization combined with combinatorial moving plane techniques, which we introduced in our earlier paper, allows to get a certain discrete monotonicity properties for the odometer function for a single source initial distributions, using which we show that the set of visited sites of the model in $\mathbb{Z}^d$ grows proportional to $n^{1/d}$, and the entire mass of the system is being distributed to an annular ring of thickness $\sim 1/m$.

We prove the existence of the scaling limit of the model generated by a single source, and fixed threshold $m$.
As we show, this limit is a ball, with the entire mass of the system being redistributed onto a narrow annular ring of thickness $\frac 1m$ near the boundary of the ball. With compactness arguments, we conclude that there is also a scaling limit for a subsequence of the odometers, if the threshold $m$ tends to infinity along with the scale of the model. That limit is spherical, with the entire mass of the system being uniformly redistributed onto the boundary of that ball. This gives a quadrature surface for a single source mass distributions.

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