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Sandpiles on the heptagonal tiling

Nikita Kalinin, Mikhail Shkolnikov

TL;DR

The work studies the abelian sandpile model on the hyperbolic heptagonal tiling Γ, focusing on perturbations of the maximal stable state and providing an explicit relaxation description via a wave_action framework. A key result is that the toppling function for a perturbation supported on P satisfies T_{φ_m^P}(v) = min(m+1−L(v), m+1−L(P)) and that a universal family α_s governs all perturbed relaxations through β_m^P = α_{L(P)} + δ_p outside P, with a constant mass_loss ratio C_m/|Γ_m| → √5. The combinatorics of Γ are encoded through precise Fibonacci-based counts of vertex types at each level, enabling exact formulas for |Γ_m|. The paper furthermore develops a wave_action mechanism that leads to a decomposition of relaxations, demonstrates a key self-similarity W_O^2 φ_m = W_O φ_{m−1}, and lays out scaling conjectures and boundary constructions for hyperbolic lattices, suggesting a program toward scaling limits on hyperbolic graphs and their abstract generalizations.

Abstract

We study perturbations of the maximal stable state in a sandpile model on the set of faces of the heptagonal tiling on the hyperbolic plane. An explicit description for relaxations of such states is given.

Sandpiles on the heptagonal tiling

TL;DR

The work studies the abelian sandpile model on the hyperbolic heptagonal tiling Γ, focusing on perturbations of the maximal stable state and providing an explicit relaxation description via a wave_action framework. A key result is that the toppling function for a perturbation supported on P satisfies T_{φ_m^P}(v) = min(m+1−L(v), m+1−L(P)) and that a universal family α_s governs all perturbed relaxations through β_m^P = α_{L(P)} + δ_p outside P, with a constant mass_loss ratio C_m/|Γ_m| → √5. The combinatorics of Γ are encoded through precise Fibonacci-based counts of vertex types at each level, enabling exact formulas for |Γ_m|. The paper furthermore develops a wave_action mechanism that leads to a decomposition of relaxations, demonstrates a key self-similarity W_O^2 φ_m = W_O φ_{m−1}, and lays out scaling conjectures and boundary constructions for hyperbolic lattices, suggesting a program toward scaling limits on hyperbolic graphs and their abstract generalizations.

Abstract

We study perturbations of the maximal stable state in a sandpile model on the set of faces of the heptagonal tiling on the hyperbolic plane. An explicit description for relaxations of such states is given.

Paper Structure

This paper contains 8 sections, 6 theorems, 20 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Combinatorics of $\Gamma$
  4. Wave action
  5. Discussion
  6. Scalings
  7. Boundaries
  8. Abstract graphs

Key Result

Proposition 1

For all $m>0$ there exist a constant $C_m>0$ such that is the same for all non-empty $P\subset\Gamma_m.$ Moreover,

Figures (6)

  • Figure 1: The triangular tiling (black edges) and the dual heptagonal tiling (white edges). The hyperbolic plane is represented by the Klein disk model.
  • Figure 2: The state $\beta_m^P$ for $m=2$ and $P=\{O\}.$ The black cell is dual to the vertex $O.$ The colors on the dual tiling represent numbers of sand at each vertex of $\Gamma_2$. The black cell represents $0,$ dark and light grey cells represent $3$ and $5$ sand grains respectively.
  • Figure 3: Two pictures of the state $\beta_6^{\{O\}}:$ in the standard Klein model and after a homothety with ratio $0.005.$
  • Figure 4: Vertices of first and second types. A type of a vertex is written in the circle, if it can be determined using the picture.
  • Figure 5: A plot of the state $W_O(\phi_m)$ (see Definitions \ref{['def_maximal']},\ref{['def_wave']}) near the boundary of $\Gamma_5.$
  • ...and 1 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • proof
  • Definition 3
  • Definition 4
  • ...and 4 more