Perturbed divisible sandpiles and quadrature surfaces

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 $\mathbb{Z}^d$ $(d\geq 2)$ which continuously deforms occupied regions of the \emph{divisible sandpile} model of Levine and Peres, 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 $m$, 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.