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Limit shape of the leaky Abelian sandpile model with multiple layers

Théo Ballu, Cédric Boutillier, Sevak Mkrtchyan, Kilian Raschel

TL;DR

This work extends the Leaky Abelian Sandpile Model to arbitrary dimensions with multiple colors and general toppling distributions, establishing a universal limit-shape phenomenon: when starting with $N$ grains at the origin and scaling by $\log N$, the final stable configuration converges to a deterministic convex limit shape described by ${\mathcal{C}}=\{\frac{u}{\Gamma^{-1}(u)\cdot u}:u\in\mathbb{S}^{d-1}\}$, where $\Gamma^{-1}$ comes from the Green-function asymptotics of a killed random walk. The authors further analyze two extreme leakiness regimes: as the leakiness $m\to\infty$ (finite-range topplings) the limit shape converges to a polytope (the dual of the walk’s step-set), and as $m\to 1$ (vanishing leakiness) the shape tends to an ellipsoid when the drift vanishes. A robust link between the sandpile’s limit shape and the Green function is developed via level-set thresholds, enabling a detailed duality description with the boundary curves being dual to spectral-radius level sets. The results unify multi-color LASMs with random-walk Green-function asymptotics, Doob transforms, and amoeba-duality concepts, yielding a rich picture of how microscopic rules shape macroscopic limit shapes in high dimensions.

Abstract

In this paper we study a triple generalization of the Leaky Abelian Sandpile Model (LASM) of Alevy and Mkrtchyan, originally analyzed in the case of the square lattice in dimension two. First, we work in any dimension. Second, each site can hold several different stacks of sand, one for each of a certain given number of different layers or colors. Third, when a stack of one color at a site topples, it can send sand not only to its nearest neighbors in equal amounts, but to all possible locations and colors, according to a fixed but arbitrary mass distribution. Stacks of different colors can topple according to different distributions and different leakiness parameters, however the toppling rule should be site-independent. We obtain three main results. First, in this generality, when the LASM is started with $N$ grains of sand in one color at the origin, the final stable configuration, after scaling down by $\log N$, converges to a limit shape as $N$ goes to infinity. Second, when the leakiness parameter converges to infinity and the toppling distribution has finite range, the limit shape converges to a polytope. Third, when the leakiness parameters converge to one, which means the leakiness disappears, the limit shape of the sandpile converges to an ellipsoid. From a technical point of view, we rely on a strong relation between the Green function for random walk and the shape of the sandpile. Finally, the limit shape exhibits interesting duality properties, which we also investigate.

Limit shape of the leaky Abelian sandpile model with multiple layers

TL;DR

This work extends the Leaky Abelian Sandpile Model to arbitrary dimensions with multiple colors and general toppling distributions, establishing a universal limit-shape phenomenon: when starting with grains at the origin and scaling by , the final stable configuration converges to a deterministic convex limit shape described by , where comes from the Green-function asymptotics of a killed random walk. The authors further analyze two extreme leakiness regimes: as the leakiness (finite-range topplings) the limit shape converges to a polytope (the dual of the walk’s step-set), and as (vanishing leakiness) the shape tends to an ellipsoid when the drift vanishes. A robust link between the sandpile’s limit shape and the Green function is developed via level-set thresholds, enabling a detailed duality description with the boundary curves being dual to spectral-radius level sets. The results unify multi-color LASMs with random-walk Green-function asymptotics, Doob transforms, and amoeba-duality concepts, yielding a rich picture of how microscopic rules shape macroscopic limit shapes in high dimensions.

Abstract

In this paper we study a triple generalization of the Leaky Abelian Sandpile Model (LASM) of Alevy and Mkrtchyan, originally analyzed in the case of the square lattice in dimension two. First, we work in any dimension. Second, each site can hold several different stacks of sand, one for each of a certain given number of different layers or colors. Third, when a stack of one color at a site topples, it can send sand not only to its nearest neighbors in equal amounts, but to all possible locations and colors, according to a fixed but arbitrary mass distribution. Stacks of different colors can topple according to different distributions and different leakiness parameters, however the toppling rule should be site-independent. We obtain three main results. First, in this generality, when the LASM is started with grains of sand in one color at the origin, the final stable configuration, after scaling down by , converges to a limit shape as goes to infinity. Second, when the leakiness parameter converges to infinity and the toppling distribution has finite range, the limit shape converges to a polytope. Third, when the leakiness parameters converge to one, which means the leakiness disappears, the limit shape of the sandpile converges to an ellipsoid. From a technical point of view, we rely on a strong relation between the Green function for random walk and the shape of the sandpile. Finally, the limit shape exhibits interesting duality properties, which we also investigate.
Paper Structure (22 sections, 47 theorems, 168 equations, 3 figures)

This paper contains 22 sections, 47 theorems, 168 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Leaky sandpile model and the associated random walk
  3. Asymptotics of the Green function
  4. The main result
  5. Literature on the Green function
  6. A homeomorphism proved by Hennequin.
  7. Second rate function in large deviation theory.
  8. Dual curves and Wulff crystals.
  9. Inverse correlation length.
  10. Anisotropic norms.
  11. Lattice Green functions and differential equations.
  12. Proof of the main asymptotics result on the Green function
  13. Odometer functionand thresholds of the Green function
  14. Limit shape when the number of grains tends to infinity
  15. Notions of convergence for sets
  16. ...and 7 more sections

Key Result

Theorem 1

In direction $u\in\mathbb S^{d-1}$, for fixed $N$, the shape of the final configuration of the sandpile lies between two radii called $r_{N, { u}}$ and $R_{N, { u}}$. We have and these limits are uniform in $u$. This means that the limit shape of the LASM is delimited by the curve and this limit holds for the Hausdorff distance (and in a stronger uniform sense, to be introduced in Definition def

Figures (3)

  • Figure 1: We consider the following sandpile model with four colors. Color $1$ sends $1$ chip to location $(0,0,0)$ to each of the colors $2,3,4$, $m=4/3$ so $1$ chip is lost. Color $2$ sends $2$ chips to color $1$, to the locations $(0,0,-1)$ and $(0,0,1)$, $m=2$, so no chip is lost. Color $3$ sends $2$ chips to color $1$, to the locations $(0,-1,0)$ and $(0,1,0)$, $m=2$, so no chip is lost. Color $4$ sends $2$ chips to color $1$, to the locations $(-1,0,0)$ and $(1,0,0)$, $m=2$, so no chip is lost. For each of the three rows, the leftmost picture shows the convex hull of the final configuration; the second picture shows a unit sphere at each point of the boundary of the visited region, the darker the sphere, the larger the height; the four right pictures show the middle horizontal slice of the final configuration in each of the $4$ colors. The first row represents the above sandpile model with initial configuration of $10^{34}$ chips. The second row represents the same sandpile with the exception that $m=10^8$ for color $1$, and with initial configuration of $10^{300}$ chips. The last row represents the same sandpile with the exception that $m=3$ for color $1$, so it is not a leaky model, and with initial configuration of $10^6$ chips.
  • Figure 2: Sets that converge to the unit ball for the Hausdorff distance, but not in the sense of Definition \ref{['def:cv_sets_limit_shape']}.
  • Figure 3: Left picture: the step set $\{(1,0),(2,-2),(0,-1),(-1,0),(1,2)\}$. Middle picture: the limit level set. Right picture: the Newton polytope associated with the step set

Theorems & Definitions (97)

  • Theorem 1: Theorem \ref{['thm:limit_shape_d_fixed']}
  • Theorem 2: Theorem \ref{['thm:limit_shape_as_convex_hull_of_returning_walks']}
  • Theorem 3: Theorem \ref{['thm:limit_shape_ellipsoid']}
  • Definition 4
  • Proposition 5
  • Lemma 6: Lem. 2.10 in Ba-24
  • proof
  • Definition 7
  • Definition 8
  • Lemma 9
  • ...and 87 more