Sandpile Solitons via Smoothing of Superharmonic Functions

Abstract

Let $F:{\mathbb {Z}}^2\rightarrow {\mathbb {Z}}$ be the pointwise minimum of several linear functions. The theory of smoothing allows us to prove that under certain conditions there exists the pointwise minimal function among all integer-valued superharmonic functions coinciding with F “at infinity”. We develop such a theory to prove existence of so-called solitons (or strings) in a 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 planar domains where solitons appear according to experiments. We prove that sandpile states, defined using our smoothing procedure, move changeless when we apply the wave operator (that is why we call them solitons), and can interact, forming triads and nodes.

Appendix A: Locally Finite Relaxations and Waves

In this section we study the relaxations and stabilizability issues. The main goal here is to establish The Least Action Principle (Proposition A.16, cf. [6]) and wave decomposition (Proposition A.26 and Corollary A.30) for locally-finite relaxations (Definition A.6) on infinite graphs. We also prove that given a finite upper bound on a toppling function of a state, there exists a relaxation sequence of this state which converges pointwise to a stable state (Lemma A.13).

The proofs are the same as in the finite case, but in the absence of references we give all the details here. Sandpiles on infinite graphs were previously considered, for example, in [1, 7, 11], but only from the distribution point of view: in their approach the relaxation (after adding a grain to a random configuration in a certain class) is locally finite almost sure with respect to a certain distribution. The ideas of this section are similar to [12].

1.1 A.1. The least action principle for locally finite relaxations, relaxability

Let $\Gamma$ be a graph with at most countable set of vertices of finite degree, $\tau :\Gamma \rightarrow {\mathbb {Z}}_{> 0}$ be a threshold function and $\gamma :\Gamma \rightarrow 2^\Gamma$ be a set-valued function such that

• $v\not \in \gamma (v),$

• if $v\in \gamma (w)$, then $w\in \gamma (v)$,

• $|\gamma (v)|\le \tau (v) \text { for all } v\in \Gamma ,$ where $|\gamma (v)|$ denotes the number of elements in the set $\gamma (v)$.

We interpret $\gamma (v)$ as the set of neighbors of a point $v\in \Gamma .$ We write $u\sim v$ instead of $u\in \gamma (v)$, because $\gamma$ induces a symmetric relation. The Laplacian $\Delta$ is the operator on the space ${\mathbb {Z}}^\Gamma =\{\phi :\Gamma \rightarrow {\mathbb {Z}}\}$ of states on $\Gamma$ given by

$$\begin{aligned} \Delta \phi (v)=-\tau (v)\phi (v)+\sum _{u\sim v}\phi (u). \end{aligned}$$

A function $\phi$ is called superharmonic if $\Delta \phi \le 0$ everywhere.

Remark A.1

Note that $|\gamma (v)|\le \tau (v) \text { for all } v\in \Gamma$ holds if and only if the function $\phi \equiv 1$ is superharmonic.

Example A.2

In our main situation, $\Gamma$ is a subset of ${\mathbb {Z}}^2$ and $|\gamma (v)|=\tau (v)=4$ for all $v\in \Gamma \setminus \partial \Gamma$. In this case we obtain the standard definition of a Laplacian on $\Gamma \setminus \partial \Gamma$:

$$\begin{aligned} \Delta \phi (v)=-4\phi (v)+\sum _{u\sim v}\phi (u). \end{aligned}$$

(A.3)

Definition A.4

For a point $v\in \Gamma$, we denote by $T_v$ the toppling operator acting on the space of states ${\mathbb {Z}}^\Gamma$. It is given by

$$\begin{aligned} T_v\phi =\phi +\Delta \delta (v), \end{aligned}$$

where $\delta (v)$ is the function on $\Gamma$ taking the value 1 at v and vanishing elsewhere.

We can think that vertices v with $\tau (v)>\gamma (v)$ are connected with the stock vertex, so the system looses sand while performing topplings at such vertices.

Definition A.5

A relaxation $\phi _{\bullet }$ of a state $\phi \in {\mathbb {Z}}^\Gamma$ is a sequence of functions $\phi _{\bullet }=\{\phi _i\}_{i \in I}$, (for $I={\mathbb {Z}}_{\ge 0}$ or $I=\{0,1,\ldots , n\}, n\in {\mathbb {Z}}_{\ge 0}$) such that $\phi _0=\phi$ and for each $k\ge 0$ there exists $v_k\in \Gamma$ such that $\phi _k(v_k)\ge \tau (v_k)$ and $\phi _{k+1}=T_{v_k}\phi _k.$ The toppling function $H_{\phi _{\bullet }}:\Gamma \rightarrow {\mathbb {Z}}_{\ge 0}\cup \{\infty \}$ of the relaxation $\phi _{\bullet }$ is given by

$$\begin{aligned} H_{\phi _{\bullet }}=\sum \limits _{i\in I}\delta ({v_i}) \in {\mathbb {Z}}_{\ge 0}\cup \{+\infty \}, \end{aligned}$$

it counts the number of topplings at every point during this relaxation. We refer to $\{v_i\}_{i\in I}$ as a relaxation sequence.

Definition A.6

A relaxation $\phi _{\bullet }$ is called locally-finite if $H_{\phi _{\bullet }}(v)$ is finite for every $v\in \Gamma .$ The result of a locally-finite relaxation is the state $\phi ^{\prime }$ given by the point-wise limit

$$\begin{aligned} \phi ^{\prime }=\phi _0+\Delta H_{\phi _{\bullet }}=\lim _{k\rightarrow \infty }\phi _k. \end{aligned}$$

Lemma A.7

Consider a locally-finite relaxation $\phi _{\bullet }$ for a state $\phi$ and a function $F:\Gamma \rightarrow {\mathbb {Z}}_{\ge 0}$ such that $\phi +\Delta F<\tau$. Then $H_{\phi _{\bullet }}(v)\le F(v)$ for all $v\in \Gamma$.

Proof

We use the notation from Definition A.5. Consider the relaxation $\phi _{\bullet }$ and the corresponding sequence of functions $H_n$ for $n=1,\ldots$ given by

$$\begin{aligned} H_n=\sum _{i=1}^n\delta ({v_i}), \end{aligned}$$

(A.8)

where $v_i$ are the points where topplings were made. Let $H_0\equiv 0$.

It suffices to show that $H_n\le F$ for every n, and $H_0\equiv 0\le F$. Suppose that $n>0$ and $H_{n-1}\le F.$ Since $H_n=H_{n-1}+\delta ({v_n}),$ it is enough to show that $H_{n-1}(v_n)<F(v_n).$ We know that $\phi _n(v_n)\ge \tau (v_n)$ and $\phi _n(v_n)=\phi _0(v_n)+\Delta H_{n-1}(v_n).$ Therefore,

$$\begin{aligned} \tau (v_n)&\le \phi _0(v_n)-\tau (v_n)H_{n-1}(v_n)+\sum _{u\sim v_n}H_{n-1}(u)\\&\le \phi _0(v_n)-\tau (v_n)H_{n-1}(v_n)+\sum _{u\sim v_n}F(u)\\&=\phi _0(v_n)+\Delta F(v_n)+\tau (v_n) \big (F(v_n)-H_{n-1}(v_n)\big ). \end{aligned}$$

Since $\phi _0(v_n)+\Delta F(v_n)< \tau (v_n)$ (by the hypothesis of the lemma) and $\tau (v_n)>0,$ we conclude that

$$\begin{aligned} 1\le F(v_n)-H_{n-1}(v_n). \end{aligned}$$

$\square$

Corollary A.9

Consider a state $\phi .$ If there exists a function $F:\Gamma \rightarrow {\mathbb {Z}}_{\ge 0}$ such that $\phi +\Delta F<\tau$, then all relaxation sequences of $\phi$ are locally finite.

Lemma A.10

Consider a state $\phi$ and the set $\Psi$ of all its relaxations $\psi _{\bullet }$. Then there exists a relaxation $\phi _{\bullet }$ of $\phi$ such that

$$\begin{aligned} H_{\phi _{\bullet }}(v)=\sup _{\psi _{\bullet }\in \Psi } H_{\psi _{\bullet }}(v), \forall v\in \Gamma . \end{aligned}$$

Proof

Consider the set $W=\{(v,k)\}\subset \Gamma \times {\mathbb {Z}}_{\ge 0}$ which contains all pairs (v, k) such that there exists a relaxation $\phi _{\bullet }^{v,k}\in \Psi$ which has k topplings at the vertex $v\in \Gamma$. Clearly, if $(v,k)\in W, k>0$ then $(v,k-1)\in W$. The set W is at most countable, so we order it as $\{(v_n,k_n)\}_{n=1,2,\ldots }$ in such a way that $(v,k-1)$ appears earlier than (v, k) for all $(v,k)\in W, k>0$.

Take any relaxation $\phi _{\bullet }$. We construct relaxations $\phi _{\bullet }^0,\phi _{\bullet }^1,\ldots$ in such a way that $\phi _{\bullet }=\phi _{\bullet }^0$, all $\phi _{\bullet }^{\ge n}$ coincide at first n topplings, and for each $n\ge 0$ the toppling function of $\phi _{\bullet }^n(v_n)$ is at least $k_n$.

Let $\phi _{\bullet }^{n-1}$ be already constructed, $n\ge 1$, we construct $\phi _{\bullet }^n$ as follows.

If the toppling function of $\phi _{\bullet }^{n-1}$ at $v_n$ is at least $k_n$, we are done. If not, take $\phi _{\bullet }^{v_{n},k_n}$ and consider its toppling functions $H_{\phi _{\bullet }^{v_{n},k_n}}^i$ as in (A.8) except that we put the bottom index to the top. Take the first i such that there exists $w\in \Gamma$ such that $H_{\phi _{\bullet }^{n-1}}(w)< H_{\phi _{\bullet }^{v_{n},k_n}}^i(w)$. Since it is the first such moment, for some j we have

$$\begin{aligned} H^j_{\phi _{\bullet }^{n-1}}(w^{\prime })\ge H_{\phi _{\bullet }^{v_{n},k_n}}^i(w^{\prime }) \end{aligned}$$

for all $w^{\prime }\sim w$. So we add to $\phi _{\bullet }^{n-1}$ the toppling at w somewhere after jth toppling, and denote the obtained relaxation sequence as $\phi _{\bullet }^{n-1}$ again. Note that by repeating this cycle of arguments a finite number of times, we will have that $\phi _{\bullet }^n(v_n)\ge k_n$. $\quad \square$

Definition A.11

A state $\phi$ is called stable if $\phi <\tau$ everywhere. A state $\phi$ is called relaxable if there exist a locally-finite relaxation $\phi _{\bullet }$ of $\phi$ such that $\phi ^{\prime }$(Definition A.6) is stable. Such a relaxation $\phi _{\bullet }$ is called stabilizing.

Corollary A.12

If $\phi$ is relaxable, then $H_{\phi _{\bullet }^1}=H_{\phi _{\bullet }^2}$ for any pair of stabilizing relaxations $\phi _{\bullet }^1$ and $\phi _{\bullet }^2$ of $\phi .$ In particular, $(\phi _{\bullet }^1)^{\circ }=(\phi _{\bullet }^2)^{\circ }$.

Proof

Applying Lemma A.7 twice, we have $H_{\phi _{\bullet }^1}\le H_{\phi _{\bullet }^2}$ and $H_{\phi _{\bullet }^1}\ge H_{\phi _{\bullet }^2}$. $\quad \square$

Lemma A.13

If all relaxations of a state $\phi$ are locally-finite, then $\phi$ is relaxable.

Proof

Consider a point $v\in \Gamma .$ We will prove that there exist $N>0$ such that $H_{\phi _{\bullet }}(v)<N$ for all relaxations $\phi _{\bullet }$ of $\phi .$ Suppose the contrary. Then there exists a sequence of relaxations $\phi ^n_{\bullet }$ such that $\lim _{n\rightarrow \infty } H_{\phi ^n_{\bullet }}(v)=\infty .$ Applying Lemma A.10 to the sequence $\phi ^n_{\bullet }$ we see that there exists a relaxation of $\phi$, that is not locally-finite.

Therefore, for any $v\in \Gamma$ there exist a relaxation $\phi ^v_{\bullet }$ such that $H_{\phi _{\bullet }}(v)\le H_{\phi ^v_{\bullet }}(v)$ for all relaxations $\phi _{\bullet }$ of $\phi .$ Applying Lemma A.10 again to the family of relaxations $\{\phi ^v_{\bullet }\}_{v\in \Gamma }$ we find a relaxation sequence $\tilde{\phi }_{\bullet }$ such that $H_{\phi _{\bullet }}(v)\le H_{\tilde{\phi }_{\bullet }}(v)$ for all relaxations $\phi _{\bullet }.$

We claim that $\tilde{\phi }_{\bullet }$ is a stabilizing relaxation. Suppose that $\phi +H_{\tilde{\phi }_{\bullet }}$ is not stable, i.e. there exists $v\in \Gamma$ such that $\phi (v) +H_{\tilde{\phi }_{\bullet }} (v)\ge \tau (v).$ Therefore, we can make an additional toppling at v after the moment when all the topplings at v and its neighbors in $\tilde{\phi }_{\bullet }$ are already made. This contradicts to the maximality of $\tilde{\phi }_{\bullet }$. $\quad \square$

Proposition A.14

A state $\phi$ is relaxable if and only if there exists a function $F:\Gamma \rightarrow {\mathbb {Z}}_{\ge 0}$ such that $\phi +\Delta F<\tau$.

Proof

If $\phi$ is relaxable then we can take F to be $H_\phi .$ On the other hand, if such F exists, then by Lemma A.7 all the relaxations of $\phi$ are locally-finite. Therefore, $\phi$ is relaxable by Lemma A.13. $\quad \square$

Definition A.15

Consider a relaxable state $\phi .$ Denote by $H_\phi$the toppling function of $\phi$, where $H_\phi$ is a toppling function of some stabilizing relaxation of $\phi .$ Define the relaxation of $\phi$ to be the state $\phi ^{\circ }=\phi +\Delta H_\phi .$

Proposition A.16

(The Least Action Principle, [6]) Let $\phi$ be a relaxable state and $F:\Gamma \rightarrow {\mathbb {Z}}_{\ge 0}$ be a function such that $\phi +\Delta F$ is stable. Then $H_\phi \le F.$ In particular, $H_\phi$ is the pointwise minimum of all such functions F.

Proof

Straightforward by Lemma A.7. $\quad \square$

Lemma A.17

Consider a stable state $\phi$ and a point $v\in \Gamma .$ Then the state $T_v\phi$ is relaxable.

Proof

Consider a function $F(z)=1-\delta (v)$ for every $z\in \Gamma$. Then $T_v\phi +\Delta F=\phi +\Delta \delta (v)+\Delta (1-\Delta \delta (v)) =\phi +\Delta 1.$ Applying Remark A.1 we see that $T_v\phi +\Delta F$ is stable. Thus, $T_v\phi$ is relaxable by Proposition A.14. $\quad \square$

1.2 A.2. Waves, their action

Sandpile waves were introduced in [10], see also [16].

Definition A.18

Let v be a point in $\Gamma .$ The wave operator $W_v$, acting on the space of the stable states on $\Gamma$, is given by

$$\begin{aligned} W_v\phi =(T_v\phi )^{\circ }. \end{aligned}$$

The wave-toppling function $H^v_\phi$ of $\phi$ at v is given by

$$\begin{aligned} H^v_\phi =\delta (v)+H_{T_v\phi }. \end{aligned}$$

(A.19)

Remark A.20

Note that if v has $\tau (v)-1$ grains and has a neighbor w with $\tau (w)-1$ grains, then the result $W_v\phi$ is non-negative everywhere.

Indeed, $T_v\phi$ has $-1$ grain at v, but w has enough grains and will topple. So, eventually, we will have non-negative amount of sand at v.

Remark A.21

It is clear that $W_v\phi =\phi +\Delta H^v_\phi .$

Corollary A.22

([26]). For any $u\in \Gamma$ the value $H^v_\phi (u)$ is either 0 or 1. Furthermore, $H^v_\phi (v)=1.$

Proof

It follows from the proof of Lemma A.17 that $H_{T_v\phi }\le 1-\delta (v)$. $\quad \square$

Lemma A.23

Suppose that $\phi$ is a stable state and v a point in $\Gamma$. If $\phi +\delta (v)$ is relaxable and not stable, then the toppling function for the wave from v is less or equal than the toppling function for a relaxation of $\phi +\delta (v)$, i.e.

$$\begin{aligned} H^v_\phi (w) \le H_{\phi +\delta (v)}(w) \ , \forall w\in \Gamma . \end{aligned}$$

Proof

It is clear that $(\phi +\delta (v))(w)=\phi (w)<\tau (w)$ for all $w\ne v$ and $(\phi +\delta (v))(v)=\tau (v).$ Therefore, $T_v$ is the first toppling in any non-trivial relaxation sequence for $\phi +\delta (v)$ and $H_{\phi +\delta (v)}(v)\ge 1.$ In particular, the function $H_{\phi +\delta (v)}-\delta (v)$ is non-negative and $H_{T_v\phi }\le H_{\phi +\delta (v)}-\delta (v)$ by Lemma A.7 since

$$\begin{aligned} T_v\phi +\Delta \big (H_{\phi +\delta (v)}-\delta (v)\big )= & {} \phi +\Delta \delta (v) +\Delta \big (H_{\phi +\delta (v)}-\delta (v)\big )=\phi + \Delta H_{\phi +\delta (v)}\\= & {} \big (\phi +\delta (v)\big )^{\circ }-\delta (v)<\tau . \end{aligned}$$

$\square$

Definition A.24

Let $\phi$ be a relaxable state, $H_\phi$ be its toppling function. Let $0\le F\le H_\phi$. The state $\phi +\Delta F$ is called a partial relaxation of $\phi$.

Lemma A.25

Consider a relaxable state $\phi$ and an integer-valued function F on $\Gamma$ such that $0 \le F\le H_\phi .$ Then the state $\phi +\Delta F$ is relaxable and

$$\begin{aligned} H_{\phi +\Delta F}=H_\phi - F. \end{aligned}$$

Proof

By Proposition A.14 the state $\phi +\Delta F$ is relaxable because

$$\begin{aligned} \phi +\Delta F+\Delta (H_\phi -F)=\phi +\Delta H_\phi =\phi ^{\circ }<\tau \end{aligned}$$

and $H_\phi -F$ is non-negative. In particular, $H_\phi -F \ge H_{\phi +\Delta F}$ by Lemma A.7. On the other hand, since $H_{\phi +\Delta F}+F\ge 0$, we have

$$\begin{aligned} \phi +\Delta (H_{\phi +\Delta F}+F)=\phi +\Delta F+\Delta H_{\phi +\Delta F}=(\phi +\Delta F)^{\circ }<\tau . \end{aligned}$$

Applying again Lemma A.7, we have $H_\phi \le H_{\phi +\Delta F}+F$. $\quad \square$

Proposition A.26

Let $\phi$ be a stable state and v be a point in $\Gamma .$ Suppose that $\phi +\delta (v)$ is relaxable. Then the relaxation of $\phi +\delta (v)$ can be decomposed into sending n waves from v, i.e.

$$\begin{aligned} (\phi +\delta (v))^{\circ }=\delta (v)+W^n_v\phi , \end{aligned}$$

where $n=H_{\phi +\delta (v)}(v)$ and $W^n_v(\phi ) = W_v(W_v(\dots (\phi ))\dots )$, nth power of $W_v$. On the level of toppling functions, this gives

$$\begin{aligned} H_{\phi +\delta (v)}=\sum _{k=0}^{n-1}H^v_{(W_v^k\phi )}. \end{aligned}$$

Added parenthesis in the subscript are for the better readability only.

Proof

Combining Lemmata A.23 and A.25 we have

$$\begin{aligned} H_{\phi +\delta (v)}=H^v_\phi +H_{(W_v\phi +\delta (v))}. \end{aligned}$$

If the state $W_v\phi +\delta (v)$ is not stable, then we can apply the same lemmata again. We complete the proof by iteration of this procedure and using Corollary A.22 (each wave has one toppling at v, therefore we have n waves). $\quad \square$

Lemma A.27

If $\phi$ is a stable state and $v_1,\ldots ,v_m$ are vertices of $\Gamma$ such that $v_i$ is adjacent to $v_{i+1}$ and $\phi (v_i)=\tau (v_i)-1$ for all $i=1,2,\ldots ,m$, then $H^{v_1}_\phi =H^{v_m}_\phi$.

Proof

It follows from the simplest case $m=2$, for which it is just a computation. $\quad \square$

Definition A.28

In a given state $\phi$, a territory is a maximal by inclusion connected component of the vertices v such that $\phi (v)=\tau (v)-1$. Given a territory ${\mathcal {T}}$, we denote by $W_{{\mathcal {T}}}$ the wave which is sent from a point in ${\mathcal {T}}$ (by Lemma A.27 it does not matter from which one).

Basically, Corollary A.22 tells us that a wave from v increases the toppling function exactly by one in the territory to which v belongs to, and by at most one in all other vertices.

Proposition A.29

Let $\phi$ be a stable stable, v be a point in $\Gamma$, and $F:\Gamma \rightarrow {\mathbb {Z}}_{\ge 0}$ be a function such that $F(v)\ge 1$ and $\phi +\Delta F$ is stable. Then $F\ge H^v_\phi$.

Proof

Similar to Lemma A.7. $\quad \square$

Corollary A.30

(Least Action Principle for waves, cf. [6]) Suppose that a state $\phi$ is stable. We send n waves from a vertex v. Let $H = \sum _{k=0}^{n-1}H^v_{(W^k\phi )}$ be the toppling function of this process. Let F be a function such that $\phi +\Delta F\ge 0, F(w)\ge 0$ for all w, and $F(v)\ge n$. Then $F(w)\ge H(w)$ for all w.

Proof

We apply Proposition A.29n times, each time decreasing F by $H^v_{W^k(\phi )}$ for $k=0,1,\ldots , n-1$. $\quad \square$