Mesoscopic concentration conjecture for localized sandpile perturbations

Establish the existence of a weak-* limit E_\rho for the rescaled density deviation N^{1/2+\alpha}(\langle 3\rangle-\phi_N) of Abelian sandpile configurations on lattices with mesh h_N=cN^{-\alpha}, for \alpha>1/2, when the empirical perturbation measures converge to a probability density \rho supported in the convex domain \Omega and disjoint from any straight boundary segments; characterize E_\rho by requiring that the solution g of \Delta g=-c^{-1}E_\rho, with Dirichlet boundary conditions, be the unique concave solution of \operatorname{MA}g=\rho with Dirichlet boundary conditions.

Background

The paper considers Abelian sandpile configurations obtained by adding N grains at perturbation sites P_N on the h_N-discretization of a compact convex planar domain \Omega and then stabilizing the maximal-density state \langle 3\rangle. The perturbation sites are assumed to have an empirical distribution converging to a prescribed probability density \rho, while the lattice mesh decreases as h_N=cN{-\alpha} with \alpha>1/2.

The conjecture concerns the next-order, microscopic description of the deviation from maximal density in a regime where the perturbations are numerous but the lattice is sufficiently fine. It predicts both weak-* convergence of the rescaled deviation and a characterization of its limit through a Poisson equation whose solution is simultaneously the Monge–Ampère solution associated with \rho.

References

At a more mathematically precise level, we put forward the following conjecture. Fix $c>0$ and $\alpha>\frac{1}{2},$ assume that $h_N$ is now expressed in terms of $N$ as $cN{-\alpha}.$ For every natural $N,$ consider states $\phi_N$ on $\Omega_{h_N}$ given by $(\langle 3 \rangle+\sum_{p\in P_N}\tilde\delta_p)\circ,$ where $P_N$ is an $N$ element subset of $\Omega_{h_N}$ such that $N{-1}\sum_{p\in P_N}\delta_p$ converges to a given probability density $\rho$ whose support is contained in $\Omega$ and disjoint from the straight segments of its boundary (if there are any). Then, there exists a weak-* limit $E_\rho$ of the rescaled deviation $N{\frac{1}{2}+\alpha}(\langle 3 \rangle-\phi_N)$ as $N\rightarrow\infty.$ Moreover, $E_\rho$ is characterized by the property that the solution $g$ of the Poisson equations $\Delta g=-c{-1}E_\rho$ serves as the unique concave solution to the Monge-Ampère equation $\operatorname{MA} g=\rho,$ where in both cases one assumes Dirichlet boundary conditions.

— Symmetry Emergence in Self-Organized Criticality  (2608.13500 - Lupercio et al., 13 Aug 2026) in Section 'From macro to micro scale'