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Sandpile solitons via smoothing of superharmonic functions

Nikita Kalinin, Mikhail Shkolnikov

TL;DR

The paper develops a smoothing theory for integer-valued superharmonic functions on $Z^2$ to construct canonical, minimal refinements $\theta_F$ of a piecewise-linear $F$ and uses these refinements to model sandpile solitons. It proves the existence (and uniqueness up to translation) of solitons for all rational slopes, and shows how solitons, triads, and nodes arise from the interaction of waves under a Least Action Principle on infinite domains. The analysis leverages reductions to cylinders, monotonicity properties, and discrete potential-theoretic tools to establish stabilization of the smoothing sequences for base functions $\Psi_{edge}$, $\Psi_{vertex}$, and $\Psi_{node}$, yielding explicit soliton profiles of the form $3+\Delta\theta_{\min(p'x+q'y,0)}$. These results connect discrete sandpile dynamics with tropical geometry and fractal pattern formation, providing a rigorous bridge between combinatorial smoothing and observed self-reproducing sandpile patterns.

Abstract

Let $F:\mathbb Z^2\to \mathbb Z$ be the pointwise minimum of several linear functions. The theory of smoothing of integer-valued superharmonic function allows us to prove that under certain conditions there exists the pointwise minimal superharmonic function which coincides with $F$ "at infinity". We develop such a theory to prove the existence of so-called solitons (or strings) in a certain sandpile model, studied by S. Caracciolo, G. Paoletti, and A. Sportiello. Thus we made a step towards understanding the phenomena of the identity in the sandpile group for a square where solitons appear according to experiments. We prove that sandpile states, defined using our smoothing procedure, move changeless when we send waves (that is why we call them solitons), and can interact, forming triads and nodes.

Sandpile solitons via smoothing of superharmonic functions

TL;DR

The paper develops a smoothing theory for integer-valued superharmonic functions on to construct canonical, minimal refinements of a piecewise-linear and uses these refinements to model sandpile solitons. It proves the existence (and uniqueness up to translation) of solitons for all rational slopes, and shows how solitons, triads, and nodes arise from the interaction of waves under a Least Action Principle on infinite domains. The analysis leverages reductions to cylinders, monotonicity properties, and discrete potential-theoretic tools to establish stabilization of the smoothing sequences for base functions , , and , yielding explicit soliton profiles of the form . These results connect discrete sandpile dynamics with tropical geometry and fractal pattern formation, providing a rigorous bridge between combinatorial smoothing and observed self-reproducing sandpile patterns.

Abstract

Let be the pointwise minimum of several linear functions. The theory of smoothing of integer-valued superharmonic function allows us to prove that under certain conditions there exists the pointwise minimal superharmonic function which coincides with "at infinity". We develop such a theory to prove the existence of so-called solitons (or strings) in a certain sandpile model, studied by S. Caracciolo, G. Paoletti, and A. Sportiello. Thus we made a step towards understanding the phenomena of the identity in the sandpile group for a square where solitons appear according to experiments. We prove that sandpile states, defined using our smoothing procedure, move changeless when we send waves (that is why we call them solitons), and can interact, forming triads and nodes.

Paper Structure

This paper contains 22 sections, 47 theorems, 79 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Sandpile patterns on $\mathbb Z^2$
  3. Superharmonic functions
  4. Sandpiles on infinite domains
  5. Acknowledgments
  6. Smoothing, its relation to waves
  7. Holeless functions
  8. Smoothing by steps
  9. Monotonicity while smoothing
  10. Discrete superharmonic integer-valued functions.
  11. Estimates on a cylinder
  12. Proof of Theorem \ref{['th_stabilfn']} for $\Psi_{\mathrm{edge}}$
  13. Classification of solitons, proof of Theorem \ref{['th_main']}
  14. Reduction to a smaller state
  15. Growth of an internal harmonic region
  16. ...and 7 more sections

Key Result

Theorem 1

For each $p,q\in\mathbb Z, \mathrm{gcd}(p,q)=1$ there exists a unique (up to a translation in $\mathbb Z^2$) movable $(p,q)$-periodic line-shaped state. Furthermore, it is $(p',q')$-movable, where $p',q'\in\mathbb Z, p'q-pq'=1$.

Figures (4)

  • Figure 1: These are local patterns for the soliton of direction $(1,3)$ and the triad made by solitons of directions $(0,-1),(1,-1),(1,2)$. White means three grains of sand, green -- two, yellow -- one, and red -- zero.
  • Figure 2:
  • Figure 3:
  • Figure 4: An illustration for the proof of Theorem \ref{['th_stabilfn']}.

Theorems & Definitions (119)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1: See a proof in Section \ref{['proof_thmain']}
  • Definition 1.4
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • ...and 109 more