On the structure of the sandpile identity element on Sierpiński gasket graphs

Robin Kaiser; Ecaterina Sava-Huss; Julia Überbacher

doi:10.58301/wzawpb66

Home Journals Meridian Journal of Mathematical SciencesVolume 11, Issue 410.58301/wzawpb66

Meridian Journal of Mathematical Sciences

Article15 November 2025

On the structure of the sandpile identity element on Sierpiński gasket graphs

Robin Kaiser^1* (unauthenticated), Ecaterina Sava-Huss², Julia Überbacher²

¹https://ror.org/02kkvpp62grid.6936.a0000 0001 2322 2966Departement of Mathematics, CIT, Technische Universität München, Boltzmannstr. 3, 85748 Garching bei München, Germany

²https://ror.org/054pv6659grid.5771.40000 0001 2151 8122Universität Innsbruck, Institut für Mathematik, Technikerstraße 13, 6020 Innsbruck, Austria

* Corresponding author.

Meridian J. Math. Sci. 2025, 11(4), 79; https://doi.org/10.58301/wzawpb66

Abstract

We consider the identity of the abelian sandpile group of finite approximation graphs of the Sierpiński gasket, and we show that the second-order term in the scaling limit converges to the path distance to the nearest corner on the Sierpiński gasket. The proof relies on a decomposition of the identity of the sandpile group into the sum of a constant function and the Laplacian of the graph distance on the approximating graphs.

Keywords:Abelian sandpilesSandpile groupIdentity sandpileSierpiński gasketGreen’s functionScaling limitSecond-order limit

Introduction

Bak et al. (1987, 1988) introduced the sandpile automaton on lattices as a system that displays self-organized criticality: it evolves into a critical state without the fine-tuning of external parameters. Dhar (1990) generalized this model to arbitrary finite graphs and noticed that the dynamics of the model are abelian, which led him to refer to it as the abelian sandpile model. Because of its relevance in physics and its interesting connections to many different fields in mathematics, the abelian sandpile model has since attracted the interest of physicists and mathematicians alike.

For a finite graph $G = (V \cup {s}, E)$ with a designated sink vertex s, a sandpile configuration is a collection of indistinguishable particles on the vertices of G. We call a vertex stable, if the number of particles is smaller than its degree, and unstable otherwise. The unstable vertices can be legally toppled by sending one particle to each neighboring vertex. During a toppling, if particles are sent to the sink, they are lost forever. By using these toppling dynamics, one can define the abelian sandpile Markov chain on the space of stable sandpile configurations in the following manner. At each time step, a particle is added at a uniformly random location, and subsequently, the configuration is stabilized by performing all possible legal topplings.

Research on the abelian sandpile model spans topics ranging from the limit shape in the single-source model—where all particles are initially placed on a single vertex and the stabilized configuration is analyzed (Pegden and Smart 2013; Levine et al. 2016; Fey et al. 2010)—through the dynamics of the sandpile Markov chain, including its mixing behaviour and stationary distribution (Jerison et al. 2019; Hough et al. 2019; Járai and Lyons 2007), to the infinite-volume limit of the stationary distribution and the size of avalanches at criticality (Járai and Werning 2014; Athreya and Járai 2004; Bhupatiraju et al. 2017; Eckmann et al. 2023).

The set of recurrent configurations for the sandpile Markov chain forms an abelian group—also known as the sandpile group or critical group—which is the focus of this note. In particular, we study the identity element of the sandpile group (shortly sandpile identity) on Sierpiński gasket approximation graphs. The sandpile identity has been examined on various graphs; for example, Le Borgne and Rossin (2002) analyzes it on rectangular grids under certain size constraints, and Melchionna (2022) investigates the identity on elliptical subsets of the square lattice. Notably, the conjecture of Dhar et al. (1995) concerning a central square of vertices with two particles each in the identity on square boxes in $Z^{2}$ remains open.

On the Sierpiński gasket, recent work has characterized and analyzed the identity element of the sandpile group. In particular, Chen and Kudler-Flam (2020) describes its structure on Sierpiński gasket graphs under various sinked boundary conditions, where one or more corner vertices are the designated sink vertices and Kaiser et al. (2024) gives a recursive characterization of the identity on the approximating graphs for normal boundary conditions, where each of the three corner vertices is connected by two edges to the sink vertex. In the current paper, we focus on the scaling limit of the sandpile identity on the approximation graphs of the Sierpiński gasket, previously studied in Kaiser and Sava-Huss (2026), where it is shown that the piecewise-constant extensions of the identities converge in the weak* sense to a constant function on the gasket. However, the structure of the identity is lost under the weak* topology. Here, we propose a different notion of limit by convolving the sandpile identity with the stopped Green’s function. This approach retains the identity’s structure and addresses a question raised in Kaiser and Sava-Huss (2026). We now present our main result. Formal definitions of the objects in the statement are deferred to Sect. 2.

Theorem 1.1

Let

{id}_{n}

be the identity of the sandpile group of the n-th approximation graph

S G_{n}

of the Sierpiński gasket SG, for

n \in N

. Denote by

g_{n}

the Green function of the simple random walk on

S G_{n}

stopped at the corners, and let G be the Green function with Dirichlet boundary conditions on the corner vertices on the Sierpiński gasket SG.

The function $g_{n} * {id}_{n}$ grows asymptotically as $5^{n}$ , and for all $x \in S G$ we have
$\begin{matrix} lim_{n \to \infty} \frac{1}{5^{n}} {(g_{n} * {id}_{n})}^{▴} (x) = 4 \int_{SG} G (x, y) d μ (y) . \end{matrix}$
The second order term of $g_{n} * {id}_{n}$ is given by the graph distance to the corners of $S G_{n}$
$lim_{n \to \infty} \frac{1}{2^{n}} (g_{n} * {id}_{n} - \frac{8}{3} \sum_{y \in V_{S G_{n}}} g_{n} (y, \cdot))^{▴} (x) = - \frac{1}{3} d (x),$
where $d (x)$ is the shortest path distance from x to the three corners in SG.
For all post-critically finite points $x \in V_{*}$ it holds that
$lim_{n \to \infty} \frac{1}{2^{n}} (g_{n} * {id}_{n} (x) - 4 \cdot 5^{n} \int_{SG} G (x, y) d μ (y)) = - \frac{1}{3} d (x) .$

Before proving Theorem 1.1, we give an informal and intuitive description of the different scaling factors and objects appearing in the statement:

The factor $5^{n}$ represents the time acceleration factor required for random walks on $S G_{n}$ to converge to a continuous Brownian motion on the Sierpinski gasket SG.
The constant 8/3 corresponds to the asymptotic average chip density of the sandpile identity element ${id}_{n}$ as $n \to \infty$ ; see Fig. 3.
Fig. 3 The identity element ${id}_{n}$ on $S G_{n}$ for $n \in {2, 3, 4, 5}$ . Blue dots are vertices with 3 chips and red dots are vertices with 2 chips (color figure online)
The scaling constant 4 emerges naturally from the standard convention used to relate the discrete and continuum Laplacians (and their inverse Green’s functions) on SG.
The measure $μ$ denotes the standard self-similar probability measure on SG.

The proof of Theorem 1.1 relies on decomposing the sandpile identity into the distance to the nearest corner vertex and a function with constant Laplacian in the interior of the gasket. More precisely, after convolving the sandpile identity with the stopped Green function on

S G_{n}

, we obtain

g_{n} * {id}_{n} = \frac{8}{3} h_{n} - \frac{1}{3} d_{n}

, where

d_{n}

denotes the distance to the closest corner, and

h_{n}

is the function with Laplacian equal to 1 at every non-corner vertex of

S G_{n}

and value 0 at the corners. Using results from analysis on fractals, we then deduce the limits stated in Theorem 1.1.

Outline. Section 2 introduces the abelian sandpile model, the Sierpiński gasket SG and its approximation graphs $S G_{n}$ , and the necessary tools from analysis on fractals used in the proof of Theorem 1.1. In Sect. 3, we establish the key ingredient: a decomposition of the sandpile identity on the approximation graphs into a function with constant Laplacian and the distance to the nearest corner. Finally, in Sect. 4, we prove Theorem 1.1: Proposition 4.3 is (I1), Proposition 4.1 is (I2), and Proposition 4.2 is (I3).

Preliminaries

Abelian sandpiles. For a comprehensive overview on the abelian sandpile model, see the survey Járai (2018). Let $G = (V \cup {s}, E)$ be a finite, connected, undirected graph with a distinguished vertex s (the sink). For $v \in V \cup {s}$ , let ${deg}_{G} (v)$ denote its degree in G, i.e., the number of its neighbors; when the graph is clear from context, we write $deg (v)$ . For vertices $v, w \in V$ , we write $v \sim w$ if they are adjacent in G (connected by an edge).

A sandpile on G is a function

σ : V \to N

that assigns to each vertex

v \in V

the number of particles at v. A vertex v is unstable if

σ (v) \geq deg (v)

; otherwise it is stable. The configuration

σ

is called stable if it is stable at every vertex. For each vertex

v \in V

, define a toppling

T_{v}

at v by removing

deg (v)

particles from v and sending one particle along each incident edge. Writing

a_{vw}

for the number of edges between v and w, the toppling redistributes particles as follows:

(T_{v} σ) (u) = \{\begin{matrix} σ (u) - deg (v), & u = v, \\ σ (u) + a_{vu}, & u \sim v, \\ σ (u), & otherwise . \end{matrix})

A toppling is legal if it results in a configuration with all nonnegative entries, i.e. if the toppled vertex v is unstable in

σ

. Any particles sent to the sink are permanently removed from the system. We view

σ

as a vector indexed by V, with each entry recording the number of particles at the corresponding vertex. With this identification, toppling operations can be expressed using the graph Laplacian of G. The Laplacian

Δ \in {Mat}_{| V \cup {s} | \times | V \cup {s} |} (Z)

is the matrix with entries

Δ_{vw} = \{\begin{matrix} - a_{vw}, & v \neq w, v \sim w, \\ deg (v), & v = w, \\ 0, & otherwise . \end{matrix})

Let

Δ^{'}

denote the reduced Laplacian of G with respect to s, obtained by restricting

Δ

to

V \times V

(i.e., deleting the row and column corresponding to the sink). Then a toppling at v is given by

T_{v} σ = σ - Δ^{'} δ_{v},

where

δ_{v}

is the vector with a 1 at index v and 0 elsewhere. This formulation immediately shows that topplings commute (the model is abelian):

T_{v} T_{w} σ = T_{w} T_{v} σ for all v, w \in V .

The stabilization

σ^{\circ}

of a sandpile

σ

is the unique stable configuration obtained by performing all legal topplings. This is well defined: there exists a sequence of vertices

v_{1}, \dots, v_{n}

—unique up to permutation—such that all topplings are legal and the resulting configuration

σ^{\circ} = T_{v_{n}} \dots T_{v_{1}} σ

is stable. See Járai (2018) for details.

Sandpile Markov chain and the sandpile group. Given sandpiles and stabilization, one defines a Markov chain on the set of stable configurations of G, known as the abelian sandpile Markov chain, as follows. Let

{(X_{i})}_{i \in N}

be i.i.d. uniformly distributed on V, and start from any stable configuration

σ_{0}

. Define the Markov chain

{(σ_{n})}_{n \in N}

by

σ_{n + 1} = {(σ_{n} + δ_{X_{n}})}^{\circ},

i.e., at time-step n, given

σ_{n}

, we add one particle to a uniformly chosen vertex in G and then we stabilize

σ_{n} + δ_{X_{n}}

. The chain

{(σ_{n})}_{n \in N}

is called the abelian sandpile Markov chain, and its set of recurrent states forms an abelian group denoted

R_{G}

, where the group operation

\oplus : R_{G} \times R_{G} \to R_{G}

is the pointwise addition followed by stabilization

σ \oplus ω = {(σ + ω)}^{\circ}

. Since

R_{G}

is an abelian group, it has a neutral element under

\oplus

: a recurrent configuration

{id}_{G}

, called the sandpile identity of G.

The Sierpiński gasket. We define Sierpiński gasket approximation graphs

S G_{n}

, for

n \in N

as in Barlow and Perkins (1988). For

x \in R^{2}

and a graph

G = (V, E)

that can be embedded in

R^{2}

, we denote by

x + G

the graph

\begin{matrix} x + G = ({x + v : v \in V}, {(x + a, x + b) : (a, b) \in E}) . \end{matrix}

For

a \in R

, we write aG for the graph

a G : = ({a v : v \in V}, {(a v, a w) : (v, w) \in E}) .

Let

u_{1} = (0, 0)

,

u_{2} = \frac{1}{2} (1, \sqrt{3})

and

u_{3} = (1, 0)

. We define

S G_{0} = (V_{S G_{0}}, E_{S G_{0}})

to be the complete graph with vertex and edge set given by

V_{S G_{0}} = {u_{1}, u_{2}, u_{3}} and E_{S G_{0}} = {{u_{1}, u_{2}}, {u_{2}, u_{3}}, {u_{1}, u_{3}}} .

Recursively, for

n \geq 1

the n-th Sierpiński gasket approximation graph

S G_{n} = (V_{S G_{n}}, E_{S G_{n}})

is

\begin{matrix} S G_{n} = \frac{1}{2} (S G_{n - 1} \cup (u_{2} + S G_{n - 1}) \cup (u_{3} + S G_{n - 1})) . \end{matrix}

Throughout this paper, we will also use the less cumbersome notation

V_{n}

for the set of vertices in the n-th iteration. Notice that

S G_{n}

is obtained from

S G_{n - 1}

by copying

S G_{n - 1}

three times and shifting one copy to the right and one copy diagonally upwards, and then rescaling the union of the three copies. Figure 1 displays the first three Sierpiński gasket graphs

S G_{0}

,

S G_{1}

and

S G_{2}

. Define the union of the vertices of all the approximation graphs

\begin{matrix} V_{*} = ⋃_{n \in N} V_{n} . \end{matrix}

The Sierpiński gasket is defined as the closure of

V_{*}

in

R^{2}

.

**Fig. 1** The first three Sierpiński gasket approximation graphs $S G_{0}$ , $S G_{1}$ and $S G_{2}$

Potential theory on the Sierpiński gasket

This subsection is devoted to introducing the necessary tools and framework from analysis on fractals—specifically analysis on the Sierpiński gasket SG—needed to prove Theorem 1.1. For a comprehensive overview, see Kigami (2001) and Strichartz (2006).

Standard self-similar measure. For each

i \in {1, 2, 3}

, define

ψ_{i} : R^{2} \to R^{2}

by

ψ_{i} (x) = \frac{1}{2} (x - u_{i}) + u_{i},

where

u_{1}, u_{2}, u_{3}

are the vertices of the previously defined equilateral triangle. For a word

w = w_{1} w_{2} \dots w_{n} \in {1, 2, 3}^{n}

of length

n \in N

, we write

ψ_{w} = ψ_{w_{n}} \circ \dots \circ ψ_{w_{1}}

. We take the

σ

-algebra of measurable sets on SG to be generated by sets of the form

{ψ_{w} (S G) : w \in ⋃_{m \geq 1} {1, 2, 3}^{m}}

. The standard self-similar measure

μ

is the unique measure with

μ (ψ_{w} (S G)) = 3^{- | w |}

, where |w| is the length of the word w. Notice that this standard self-similar measure is up to scalar multiples a

{log}_{2} (3)

-dimensional Hausdorff measure on the Sierpiński gasket with respect to the Euclidean metric. See also Chapter 1.2 in Strichartz (2006, pp. 5–9) for more details.

Green function on

S G_{n}

. Let

{(X_{t}^{n})}_{t \in N}

denote the simple random walk on

S G_{n}

, i.e., the Markov chain with transition probabilities given by

\begin{matrix} p (x, y) = \{\begin{matrix} \frac{1}{4}, if x \underset{n}{\sim} y, x \neq u_{i} \\ \frac{1}{2}, if x \underset{n}{\sim} y, x = u_{i} \end{matrix}) \end{matrix}

for

i \in {1, 2, 3}

, where

x \underset{n}{\sim} y

means that

x, y \in V_{S G_{n}}

are neighbours in

S G_{n}

. For

x \in V_{S G_{n}}

, denote by

E_{x}

the expectation with respect to the probability measure

P (\cdot | X_{0}^{n} = x)

. The Green function

g_{n} (x, y)

on

S G_{n}

is the expected number of visits to y by the walk started at x before it hits one of the three corner vertices

u_{1}, u_{2}, u_{3}

:

g_{n} (x, y) = E_{x} [\sum_{t = 0}^{τ_{n} - 1}, 1_{{X_{t}^{n} = y}}], where τ_{n} = inf \{t \geq 0 : X_{t}^{n} \in {u_{1}, u_{2}, u_{3}}\} .

If we write

Δ_{n}

for the graph Laplacian on

S G_{n}

, then it holds that

Δ_{n} g_{n} (x, \cdot) (y) = δ_{x} (y), for all x, y \in S G_{n} \ {u_{1}, u_{2}, u_{3}} .

Piecewise constant continuation to SG. For

x \in S G

, define

x_{n} \in V_{S G_{n}}

as

x_{n} = argmin {d (x, y) : y \in S G_{n}}

. For uniqueness, if there are several minimizers, choose the one furthest to the right. For a function

f : V_{S G_{n}} \to R

we define the piecewise constant continuation of f to SG by

f^{▴} : S G \to R, f^{▴} (x) = f (x_{n})

.

Laplacian operator on SG. The Laplacian operator on the Sierpiński gasket SG, denoted by

Δ

, can be defined via a weak formulation using an energy form on SG. For

n \in N

, we define the energy form

E_{n}

on

S G_{n}

as

E_{n} (f, g) = (\frac{5}{3})^{n} \sum_{x \underset{n}{\sim} y} (f (x) - f (y)) (g (x) - g (y)),

where

f, g : S G \to R

. Observe that the 5/3 factor in the energy functional corresponds to the increase in minimal energy when a function on the n-th gasket is extended to the

(n + 1)

-st. For details, see Strichartz (2006, Section 1.3). Using these finite iteration energy forms

E_{n}

we define an energy form on SG via

E (f, g) = lim_{n \to \infty} E_{n} {(f |}_{V_{S G_{n}}}, g |_{V_{S G_{n}}}) .

Note that the limit is not necessarily finite. So, we also define the domain of the energy functional as

dom (E) : = {f : S G \to R ∣ E (f, f) < \infty} .

With this energy form, we define a weak Laplacian on SG. For

u \in dom (E)

and continuous

f : S G \to R

, we write

u \in dom (Δ)

and

Δ u = f

if and only if

E (u, v) = - \int_{SG} f v d μ for all v \in dom (E) .

Green function on SG. For Theorem 1.1, we must specify the function on SG that arises as the scaling limit of the discrete Green functions

g_{n}

defined above. Specifically, we define the Green function on the Sierpiński gasket SG, i.e., the integral kernel of the inverse Laplacian on SG. We define the Green function subject to Dirichlet boundary conditions on SG via harmonic splines as explained in Strichartz (2006, Section 2.6.) by

G (x, y) : = lim_{M \to \infty} \sum_{m = 0}^{M} \sum_{z, z^{'} \in V_{S G_{m + 1}} \ V_{S G_{m}}} g (z, z^{'}) Ψ_{z}^{m + 1} (x) Ψ_{z^{'}}^{m + 1} (y),

where

g (z, z^{'}) = \{\begin{matrix} \frac{9}{50} (\frac{3}{5})^{m}, & z = z^{'} and z \in V_{S G_{m + 1}} \ V_{S G_{m}}, \\ \frac{3}{50} (\frac{3}{5})^{m}, & z \neq z^{'}, z, z^{'} \in V_{S G_{m + 1}} \ V_{S G_{m}}, and z, z^{'} \in ψ_{w} (S G) for some | w | = m, \end{matrix})

and, for

z \in V_{S G_{m}}

, the level-m harmonic spline

Ψ_{z}^{m}

is the unique piecewise harmonic function on SG satisfying

Ψ_{z}^{m} (x) = \{\begin{matrix} 1, & x = z, \\ 0, & x \in V_{m} \ {z} . \end{matrix})

A key feature of the Green function on SG is that, up to the scaling factor

{(3 / 5)}^{n}

, it matches the stopped Green function

g_{n}

on the vertices of

S G_{n}

.

Lemma 2.1

For all $n \in N$ and all $x, y \in V_{n}$ it holds ${(\frac{3}{5})}^{n} g_{n} (x, y) = G (x, y) .$

For a proof see, for instance, Lemma 2.4 in Freiberg et al. (2024).

Auxiliary results

In the proofs of the scaling limit in Sect. 4, we need several technical lemmas. These include the mean-value property of harmonic functions on SG and properties of the Green function G on the SG, specifically the variation of the Green function within a given n-cell.

Lemma 2.2

Let

h : S G \to R

be a harmonic function on

S G \ V_{0}

. Then for every

m, n \in N

with

m > n

and

x \in V_{n} \ V_{0}

it holds that

\frac{1}{μ (B (x, 2^{- m}))} \int_{B (x, 2^{- m})} h (y) d μ (y) = h (x) .

Proof

The statement follows from the well-known fact (Strichartz 2006, Section 1.3.) that a harmonic function h on an n-cell of SG is determined by its values at the three corner vertices via the 1/5–2/5 rule. More precisely, if

x_{1}, x_{2}, x_{3} \in V_{n}

are the corner vertices and

y_{1}, y_{2}, y_{3} \in V_{n + 1} \ V_{n}

are the corresponding cut points (each

y_{i}

opposite

x_{i}

), the rule gives

h = h (x_{1}) ψ_{1} + h (x_{2}) ψ_{2} + h (x_{3}) ψ_{3}

on this n-cell, where

ψ_{i}

is the unique harmonic spline with

ψ_{i} (x_{j}) = δ_{ij}

, and the algorithm gives

\begin{matrix} ψ_{1} (y_{1}) = \frac{1}{5}, ψ_{1} (y_{2}) = ψ_{1} (y_{3}) = \frac{2}{5}, \end{matrix}

1

The other values of

ψ_{i} (y_{j})

are found via dihedral symmetry. Consequently h is discrete harmonic:

\begin{matrix} h (y_{1}) = \frac{1}{4} (h (x_{2}) + h (x_{3}) + h (y_{2}) + h (y_{3})) . \end{matrix}

2

Moreover, since

\sum_{i = 1}^{3} ψ_{i} \equiv 1

and

μ (m -cell) = 3^{- m}

for every

m \geq n

we deduce by symmetry that for every

(n + 1)

-subcell

T_{n + 1}

contained in the n-cell,

\int_{T_{n + 1}} ψ_{i} d μ = 3^{- (n + 2)}

for

i \in {1, 2, 3}

. We now establish the mean-value property of h over the ball

B (y_{1}, 2^{- (n + 1)})

:

\begin{matrix} \frac{1}{μ (B (y_{1}, 2^{- (n + 1)}))} \int_{B (y_{1}, 2^{- (n + 1)})} h d μ = h (y_{1}) . \end{matrix}

3

Using (1) and the above integrals of

ψ_{i}

, we get

\begin{matrix} \int_{B (y_{1}, 2^{- (n + 1)})} h d μ & = (h (y_{1}) + h (x_{3}) + h (y_{2})) \cdot 3^{- (n + 2)} + (h (y_{1}) + h (x_{2}) + h (y_{3})) \cdot 3^{- (n + 2)} \\ = \frac{1}{3} \cdot 3^{- (n + 1)} \cdot (h (x_{2}) + h (x_{3}) + h (y_{2}) + h (y_{3}) + 2 h (y_{1})) \\ = \frac{1}{3} \cdot 3^{- (n + 1)} \cdot 6 h (y_{1}), \end{matrix}

where we used (2) in the last equation. Dividing the last expression by

μ (B (y_{1}, 2^{- (n + 1)})) = 2 \cdot 3^{- (n + 1)}

gives (3). In (3) we apply the self-similar cell structure of SG to deduce the claimed mean-value property.

□

The next result relates the Green function on SG to the effective resistance. The definition of the effective resistance metric is standard, and can for example be found in Strichartz (2006, Section 1.6.).

Proposition 2.1

Let

R_{eff} (A, B)

be the effective resistance between subsets

A, B \subset S G

, subject to Dirichlet boundary conditions on

V_{0}

. Let G(x, y) be the Green’s function on SG subject to Dirichlet boundary conditions on

V_{0}

. Then the following holds:

For every $x \in S G$ , $G (x, x) = R_{eff} (x, V_{0})$ .
For every $x, y \in S G$ , $G (x, y) = \frac{1}{2} (G (x, x) + G (y, y) - R_{eff} (x, y))$ .
For every $x, y \in S G$ we have $G (x, x) \geq G (x, y)$ .

Proof

These follow from the general theory of Green functions on resistance forms, which is detailed in Kigami (1015, Section 4). For proofs, see Theorems 4.1 and Theorem 4.3 from Kigami (1015). $□$

Lemma 2.3

For every

x, y, z \in S G

we have

| G (x, y) - G (x, z) | \leq R_{eff} (y, z),

and consequently

0 \leq G (x, x) - G (x, y) \leq R_{eff} (x, y) .

Proof

Using (1) and (2) in Proposition 2.1, we obtain

\begin{matrix} G (x, y) - G (x, z) & = \frac{1}{2} [G (y, y) - G (z, z) + R_{eff} (x, z) - R_{eff} (x, y)] \\ = \frac{1}{2} [R_{eff} (y, V_{0}) - R_{eff} (z, V_{0}) + R_{eff} (x, z) - R_{eff} (x, y)] . \end{matrix}

Note that

V_{0}

is grounded and hence can be collapsed to a single vertex. Meanwhile, the resistance metric satisfies the triangle inequality, so

\begin{matrix} | G (x, y) - G (x, z) | & \leq \frac{1}{2} (| R_{eff} (y, V_{0}) - R_{eff} (z, V_{0}) | + | R_{eff} (x, z) - R_{eff} (x, y) |) \\ \leq \frac{1}{2} (R_{eff} (y, z) + R_{eff} (y, z)) = R_{eff} (y, z), \end{matrix}

proving the first inequality. This together with (3) from Proposition 3 implies the second part.

□

Lemma 2.4

There exists a constant

C > 0

such that the following hold.

For every $x, y \in S G$ belonging to the same n-cell, we have
$0 \leq G (x, x) - G (x, y) \leq C {(\frac{3}{5})}^{n},$
For every $x \in S G$ , $n \in N$ and $y \in S G$ with $d (y, V_{0}) \leq 2^{- n}$ we have
$0 \leq G (x, y) \leq C {(\frac{3}{5})}^{n} .$

Proof

For the first claim, using Lemma 2.3 it suffices to show that there exists $C > 0$ such that for every $x, y \in S G$ belonging to the same n-cell, we have $R_{eff} (x, y) \leq C {(\frac{3}{5})}^{n}$ . This is the well-known resistance upper bound (Strichartz 2006, Section 1.6). For the second claim, using the third and then the first item in Proposition 2.1, we obtain $G (x, y) \leq G (y, y) = R_{eff} (y, V_{0})$ . Since y shares the same n-cell with one of the corner vertices in $V_{0}$ , applying the aforementioned resistance upper bound yields $R_{eff} (y, V_{0}) \leq C {(\frac{3}{5})}^{n}$ . $□$

Sandpiles on the Sierpiński gasket graphs

Boundary conditions on

S G_{n}

. For the abelian sandpile Markov chain on

S G_{n}

, we adopt the standard “normal” boundary conditions used in the literature (e.g., Kaiser and Sava-Huss 2026): introduce a sink vertex

s \notin V_{S G_{n}}

and connect it by two edges to the corner vertices of

S G_{n}

, so every vertex in

V_{S G_{n}}

has degree 4 (see Fig.

2

). Throughout, “sandpile group on

S G_{n}

” refers to the set of recurrent configurations of the sandpile Markov chain with this boundary, denoted

R_{n}

, and we write

{id}_{n}

for its identity element.

**Fig. 2** The Sierpiński gasket graphs $S G_{0}$ , $S G_{1}$ and $S G_{2}$ with normal boundary conditions

Sandpile identity on

S G_{n}

. Define the sandpile configuration

M_{1} (x, y, z)

on

S G_{1}

as

where the corner values

x, y, z \in N

are arbitrary. Iteratively define the configuration

M_{n + 1} (x, y, z)

In Kaiser et al. (2024), the authors prove in Theorem 3.2 that for any

n \geq 1

, the identity element of

S G_{n + 1}

with normal boundary conditions is given by

where

M_{n}^{+}

(respectively

M_{n}^{-}

) denotes the sandpile configuration obtained from

M_{n}

by rotating

S G_{n}

counterclockwise (respectively clockwise) by

120^{\circ}

. The sandpile identity elements on the Sierpiński gaskets

S G_{n}

for

n \in {2, 3, 4, 5}

are displayed in Fig. 3.

Decomposition of the sandpile identity

In this section, we establish a decomposition of the sandpile identity needed for proving the main theorem. For

n \in N

, define

I_{n} : V_{S G_{n}} \to R

by

I_{n} (x) = (g_{n} * {id}_{n}) (x) : = \sum_{y \in V_{S G_{n}}} g_{n} (x, y) {id}_{n} (y) .

We have

Δ_{n} I_{n} = {id}_{n}

on

V_{S G_{n}} \ {u_{1}, u_{2}, u_{3}}

and

I_{n} (u_{i}) = 0

for each

i \in {1, 2, 3}

; that is,

I_{n}

vanishes at the corner vertices of

S G_{n}

. We also define

h_{n} : V_{S G_{n}} \to R

by

\begin{matrix} h_{n} (x) = \sum_{y \in V_{S G_{n}}} g_{n} (x, y) . \end{matrix}

4

It follows that

Δ_{n} h_{n} (x) = 1

for all

x \in V_{S G_{n}} \ {u_{1}, u_{2}, u_{3}}

, and

h_{n}

vanishes at the corner vertices of

S G_{n}

. Next, define the distance to the corners by

\begin{matrix} d_{n} : V_{S G_{n}} \to R, d_{n} (x) = min {d_{S G_{n}} (x, u_{i}) : i \in {1, 2, 3}}, \end{matrix}

5

where

d_{S G_{n}}

denotes the graph distance in

S G_{n}

. In the next proposition, we show that

I_{n}

can be expressed as a linear combination of the functions

d_{n}

and

h_{n}

.

Proposition 3.1

For

n \geq 2

and

I_{n} = g_{n} * {id}_{n}

, it holds that

I_{n} = - \frac{1}{3} d_{n} + \frac{8}{3} h_{n} .

Proof

We write

F_{n} : = - \frac{1}{3} d_{n} + \frac{8}{3} h_{n}

and we first show that

Δ_{n} F_{n} (x) = {id}_{n} (x) for all x \in V_{S G_{n}} \ {u_{1}, u_{2}, u_{3}} .

Since the functions

d_{n}

and

h_{n}

are rotationally symmetric, it suffices to consider

Δ_{n} F_{n}

only on the top triangle

△_{n}^{2}

. Special attention must be paid to the cutpoints whose neighborhoods intersect more than one of the three subtriangles, since the Laplacian at a vertex depends on the values of the function on its neighboring vertices. We write

▵_{n}^{2}

for the vertices of

S G_{n}

in the upper triangle

▵_{n}^{2} = ψ_{2} (V_{S G_{n - 1}} \ {u_{1}, u_{2}, u_{3}})

and similarly

▵_{n}^{1} = ψ_{1} (V_{S G_{n - 1}} \ {u_{1}, u_{2}, u_{3}}), ▵_{n}^{3} = ψ_{3} (V_{S G_{n - 1}} \ {u_{1}, u_{2}, u_{3}}) .

We write $p_{1}, p_{2}$ and $p_{3}$ for the three cutpoints, so $p_{1} = \frac{1}{2} u_{2}$ , $p_{2} = \frac{1}{2} u_{3}$ , $p_{3} = p_{1} + p_{2}$ ; see Fig. 4. Below, for readability reasons, we suppress the subscript n of the Laplacians in the calculations. We prove the statement by induction on the iteration of the gasket graphs; let $n = 2$ .

The graph distance from the top vertex

u_{2}

and its Laplacian are given by

and the white circles placed at the cutpoints indicate values that we compute later. To emphasize that we work on

△_{2}^{2}

, we have colored the top vertex

u_{2}

black. Since the Laplacian of

h_{2}

is 1 in all non-corner vertices, the Laplacian of

F_{2}

on

△_{2}^{2}

can be computed as

For the values at the cutpoints we have

This proves the statement for $n = 2$ .

Next we assume that $Δ F_{n} {|_{S G_{n} \ {u_{1}, u_{2}, u_{3}}} = {id}_{n} |}_{S G_{n} \ {u_{1}, u_{2}, u_{3}}}$ holds $n \in N$ , and we show that the statement holds for $n + 1$ . We restrict again our attention to the upper subtriangle $▵_{n + 1}^{2}$ . Within $▵_{n + 1}^{2}$ , the distance function $d_{n + 1}$ coincides with the graph distance to the top corner vertex $u_{2}$ . Thus, on $▵_{n + 1}^{2}$ , the distance from the top corner can be depicted as in Fig. 5.

Since the Laplacian at x depends solely on the function’s values at the neighbors of x, for all vertices in the blue triangle of Fig. 5 that have no neighbors in the red or green triangles, the claim follows directly from the induction hypothesis. Next, consider vertices lying in the green or red triangles that have no neighbors outside their respective triangles. Since within both the green and red regions the distance is uniformly shifted by the constant

2^{n - 1}

, the claim again follows from the induction hypothesis. Therefore, for any vertex x in the red, green, or blue triangle whose neighbors all lie in the same subtriangle as x, we have

Δ F_{n + 1} (x) = {id}_{n + 1} (x)

. Now we examine the values of the Laplacian at the cutpoints

c_{2}^{'}, c_{2}^{''}

and m in

▵_{n + 1}^{2}

as shown in Fig. 5. For

c_{2}^{'}

we have

\begin{matrix} Δ d_{S G_{n + 1}} (u_{2}, c_{2}^{'}) & = \sum_{y \sim c_{2}^{'}} d_{S G_{n + 1}} (u_{2}, c_{2}^{'}) - d_{S G_{n + 1}} (u_{2}, y) \\ = 4 \cdot 2^{n - 1} - ((2^{n - 1} - 1) + 2^{n - 1} + 2 \cdot (2^{n - 1} + 1)) = - 1 . \end{matrix}

By symmetry, for

c_{2}^{''}

it holds that

Δ d_{S G_{n + 1}} (u_{2}, c_{2}^{''}) = - 1

, and for m similar computations lead to

\begin{matrix} Δ d_{S G_{n + 1}} (u_{2}, m) = 4 \cdot 2^{n} - (2 \cdot (2^{n} - 1) + 2 \cdot 2^{n}) = 2 . \end{matrix}

Hence, on

▵_{n + 1}^{2}

we have

Δ F_{n + 1} {|_{▵_{n + 1}^{2}} = {id}_{n + 1} |}_{▵_{n + 1}^{2}}

, and by rotational symmetry, the same holds on

▵_{n + 1}^{1}

and

▵_{n + 1}^{3}

. It remains to address the cutpoints

p_{1}, p_{2}, p_{3}

of

S G_{n + 1}

; see Fig. 4 for the cutpoints of

S G_{n}

. By symmetry, the values of

Δ d_{n + 1} (\cdot)

at these vertices agree and are equal to

\begin{matrix} Δ d_{n + 1} (p_{1}) & = \sum_{v \sim p_{1}} d_{n + 1} (p_{1}) - d_{n + 1} (v) = 4 \cdot 2^{n} - (2 \cdot 2^{n} + 2 \cdot (2^{n} - 1)) = 2 . \end{matrix}

This completes the argument, yielding

Δ_{n} F_{n} (x) = {id}_{n} (x)

for all

n \in N

and all

x \in V_{S G_{n}} \ {u_{1}, u_{2}, u_{3}}

. Define

T_{n} = F_{n} - I_{n}

. Then

Δ_{n} T_{n} (x) = 0

for all

x \in V_{S G_{n}} \ {u_{1}, u_{2}, u_{3}}

and

T_{n} (u_{i}) = 0

for each

i \in {1, 2, 3}

. By the maximum (–minimum) principle for harmonic functions,

T_{n} (x) = 0

for all

x \in V_{S G_{n}}

, hence

I_{n} = F_{n}

, completing the proof.

□

**Fig. 5** The values of the graph distance function in the subtriangle $△_{n + 1}^{2}$

Scaling limit of the sandpile identity

We prove the three parts of Theorem 1.1 separately, beginning with (I2). This follows immediately from the decomposition of the sandpile identity established in Proposition 3.1.

Let

d_{SG} (x, y)

denote the path distance between points

x, y \in S G

, which can be obtained from the graph distance on

S G_{n}

via

d_{SG} (x, y) = lim_{n \to \infty} \frac{1}{2^{n}} d_{S G_{n}} (x_{n}, y_{n}),

where

x_{n}

and

y_{n}

are the vertices in

S G_{n}

closest to x and y respectively. Write

d (x) = min {d_{SG} (x, u_{1}), d_{SG} (x, u_{2}), d_{SG} (x, u_{3})} .

Proposition 4.1

It holds that

lim_{n \to \infty} \frac{1}{2^{n}} {(g_{n} * {id}_{n} - \frac{8}{3} h_{n})}^{▴} (x) = - \frac{1}{3} d (x) .

Proof

In view of Proposition 3.1

g_{n} * {id}_{n} (x) - \frac{8}{3} h_{n} (x) = - \frac{1}{3} d_{n} (x),

for all

x \in V_{S G_{n}}

, from which the claim follows by using

lim_{n \to \infty} \frac{1}{2^{n}} {(g_{n} * {id}_{n} - \frac{8}{3} h_{n})}^{▴} (x) = lim_{n \to \infty} - \frac{1}{3 \cdot 2^{n}} d_{n}^{▴} (x) = - \frac{1}{3} d (x) .

□

We now address statement (I3) of Theorem 1.1. The first step is to establish that the functions $h_{n}$ provide a good approximation to the integral of the Green function G over the Sierpiński gasket.

Lemma 4.1

There exists a constant

c_{*} > 0

such that for all

x \in V_{*}

, it holds that

\underset{n \to \infty}{lim sup} |3 \cdot 5^{n} \int_{SG} G (x, y) d μ (y) - 2 \sum_{y \in V_{n}} g_{n} (x, y)| \leq c_{*} .

Proof

First note that if $x \in V_{0}$ , then $G (x, y) = 0$ for all $y \in S G$ , and $g_{n} (x, y) = 0$ for all $n \in N$ and all $y \in V_{n}$ , making the lemma vacuously true.

Therefore we fix

x \in V_{*} \ V_{0}

. By definition,

x \in V_{n}

for all sufficiently large

n \in N

. Also, given

n \in N

we may partition SG into balls of diameter

2^{- (n + 1)}

in the Euclidean distance centered at each

z \in V_{n}

,

S G = ⋃_{z \in V_{n}} B (z, 2^{- (n + 1)})

. If

z \in V_{n} \ V_{0}

, then the closed ball

B (z, 2^{- (n + 1)})

consists of two

(n + 1)

-cells adjoined at z, and if

z \in V_{0}

, then

B (z, 2^{- (n + 1)})

is the single

(n + 1)

-cell containing the corner vertex z. As a result, under the standard self-similar probability measure

μ

, the mass assigned to each ball is

\begin{matrix} μ (B (z, 2^{- (n + 1)})) = \{\begin{matrix} 2 \cdot 3^{- (n + 1)}, & if z \in V_{n} \ V_{0} \\ 3^{- (n + 1)}, & if z \in V_{0} \end{matrix}) \end{matrix}

6

Note that while two different balls may intersect, the set of their intersection has

μ

-measure 0. We can thus express the integral of the Green’s function as the sum of contributions from each ball of the said partition

\begin{matrix} \int_{SG} G (x, y) d μ (y) & = \sum_{z \in V_{n}} \int_{B (z, 2^{- (n + 1)})} G (x, y) d μ (y) \\ = \sum_{z \in V_{n} \ V_{0}} \int_{B (z, 2^{- (n + 1)})} G (x, y) d μ (y) + \sum_{z \in V_{0}} \int_{B (z, 2^{- (n + 1)})} G (x, y) d μ (y) \\ = : I_{bulk} + I_{boundary} . \end{matrix}

To analyze

I_{bulk}

, by the harmonic spline construction of the Green’s function, the function

G (x, \cdot)

is harmonic on

B (z, 2^{- (n + 1)})

as long as

z \neq x

and

z \notin V_{0}

. Applying Lemma 2.2, we get

\begin{matrix} \frac{1}{μ (B (z, 2^{- (n + 1)}))} \int_{B (z, 2^{- (n + 1)})} G (x, y) d μ (y) = G (x, z) for every z \in V_{n} \ V_{0}, z \neq x . \end{matrix}

7

which together with Eqs. (6) and (7) yields

\begin{matrix} I_{bulk} & = 2 \cdot 3^{- (n + 1)} \sum_{z \in V_{n} \ V_{0}, z \neq x} G (x, z) + \int_{B (x, 2^{- (n + 1)})} G (x, y) d μ (y) \\ = 2 \cdot 3^{- (n + 1)} \sum_{z \in V_{n}} G (x, z) - 2 \cdot 3^{- (n + 1)} G (x, x) + \int_{B (x, 2^{- (n + 1)})} G (x, y) d μ (y) \\ = \frac{2}{3} \cdot \frac{1}{5^{n}} \sum_{z \in V_{n}} g_{n} (x, z) + \int_{B (x, 2^{- (n + 1)})} (G (x, y) - G (x, x)) d μ (y) . \end{matrix}

Above, we used the fact that

G (x, z) = 0

if one of x and z belongs to

V_{0}

, as well as Lemma 2.1 relating G to the discrete Green function. Turning now to the main expression in the Lemma, we can write

\begin{matrix} \begin{matrix} 3 \cdot 5^{n} & \int_{SG} G (x, y) d μ (y) - 2 \sum_{y \in V_{n}} g_{n} (x, y) = 3 \cdot 5^{n} (I_{bulk} + I_{boundary}) - 2 \sum_{y \in V_{n}} g_{n} (x, y) \\ = 3 \cdot 5^{n} \int_{B (x, 2^{- (n + 1)})} (G (x, y) - G (x, x)) d μ (y) + 3 \cdot 5^{n} \sum_{z \in V_{0}} \int_{B (z, 2^{- (n + 1)})} G (x, y) d μ (y) . \end{matrix} \end{matrix}

8

We claim that this expression is uniformly bounded in

n \in N

and in

x \in V_{*}

by a universal constant. To prove this, we need two results concerning the Green’s function, each of which is used to bound each of the two terms in (8). The first result bounds the variation of the Green function on the balls

B (x, 2^{- (n + 1})

for all

x \in V_{*} \ V_{0}

,

n \in N

and

y \in B (x, 2^{- (n + 1)})

and this is part (1) in Lemma 2.4, showing that the first term in Eq. (8) is negative. The second needed result bounds the decay of the Green function near the boundary

V_{0}

and this is part (2) of Lemma 2.4, showing that the second term in (8) is positive. Therefore the absolute value of (8) is bounded above by

\begin{matrix} max {3 \cdot 5^{n} \int_{B (x, 2^{- (n + 1)})} (G (x, x) - G (x, y)) d μ (y), 3 \cdot 5^{n} \sum_{z \in V_{0}} \int_{B (z, 2^{- (n + 1)})} G (x, y) d μ (y)} \\ \leq max {3 \cdot 5^{n} \cdot 2 \cdot 3^{- (n + 1)} \cdot C {(\frac{3}{5})}^{n + 1}, 3 \cdot 5^{n} \cdot 3 \cdot 3^{- (n + 1)} \cdot C {(\frac{3}{5})}^{n + 1}} \\ = max {\frac{6}{5} C, \frac{9}{5} C} = \frac{9}{5} C, \end{matrix}

thereby proving the above claim and hence the Lemma.

□

We now prove (I3) in Theorem 1.1.

Proposition 4.2

For all

x \in V_{*}

it holds that

lim_{n \to \infty} \frac{1}{2^{n}} (g_{n} * {id}_{n} (x) - 4 \cdot 5^{n} \int_{SG} G (x, y) d μ (y)) = - \frac{1}{3} d (x) .

Proof

Let

x \in V_{*}

, then by Proposition 3.1

\begin{matrix} \frac{1}{2^{n}} (g_{n} * {id}_{n} (x) - 4 \cdot 5^{n} \int_{SG} G (x, y) d μ (y)) \\ = - \frac{1}{3 \cdot 2^{n}} d_{n} (x) + \frac{4}{3 \cdot 2^{n}} (2 h_{n} (x) - 3 \cdot 5^{n} \int_{SG} G (x, y) d μ (y)) . \end{matrix}

Lemma 4.1 gives

\frac{4}{3 \cdot 2^{n}} | 2 h_{n} (x) - 3 \cdot 5^{n} \int_{SG} G (x, y) d μ (y) | \leq c_{*} \frac{4}{3 \cdot 2^{n}} \overset{n \to \infty}{\to} 0,

which together with

{lim}_{n} 2^{- n} d_{S G_{n}} (x) = d (x)

yield the claim.

□

Using our previous result, we can now directly prove (I1) in Theorem 1.1 by using the already established (I3).

Proposition 4.3

The function

g_{n} * {id}_{n}

grows asymptotically as

5^{n}

, and for all

x \in S G

it holds that

\begin{matrix} lim_{n \to \infty} \frac{1}{5^{n}} {(g_{n} * {id}_{n})}^{▴} (x) = 4 \int_{SG} G (x, y) d μ (y) . \end{matrix}

Proof

Recall for every

n \in N

the definition of the piecewise-constant continuatione of

u^{▴} : S G \to R, u^{▴} (x) = u (x_{n})

, where

x_{n} \in V_{n}

is the closest point to

x \in S G

in the n-th iteration. Let us introduce the shorthand notation

β_{n} : S G \to R

given by

β_{n} (x) : = \frac{1}{5^{n}} {(g_{n} * i d_{n})}^{▴} (x) - 4 \int_{SG} G (x, y) d μ (y),

which can be rewritten as

\begin{matrix} β_{n} (x) & = (\frac{1}{5^{n}} {(g_{n} * i d_{n})}^{▴} (x) - 4 \int_{SG} {(G (\cdot, y))}^{▴} (x) d μ (y)) \\ + 4 \int_{SG} ({(G (\cdot, y))}^{▴} (x) - G (x, y)) d μ (y) \\ = : D_{n} (x) + D_{n}^{'} (x) . \end{matrix}

Using Proposition 4.2 and

{sup}_{x \in S G} d (x) = \frac{1}{2}

we get

\begin{matrix} \underset{n \to \infty}{lim sup} | D_{n} (x) | & \leq \underset{n \to \infty}{lim sup} \frac{1}{2^{n}} |{(g_{n} * i d_{n})}^{▴} (x) - 4 \cdot 5^{n} \int_{SG} {(G (\cdot, y))}^{▴} (x) d μ (y)| \\ \cdot \underset{n \to \infty}{lim sup} \frac{2^{n}}{5^{n}} = 0 . \end{matrix}

On the other hand, by Lemma 2.3, for every

x \in S G

we have

| {(G (\cdot, y))}^{▴} (x) - G (x, y) | = | G (x_{n}, y) - G (x, y) | \leq R_{eff} (x_{n}, x) \overset{n \to \infty}{\to} 0,

which together with the dominated convergence theorem yields

{lim}_{n \to \infty} D_{n}^{'} (x) = 0

. Hence it holds

{lim}_{n \to \infty} β_{n} (x) = 0

and this completes the proof.

□

Remark 1

In this work, we examined the sequence of identity elements in the sandpile group on the Sierpiński gasket approximation graphs and showed that, after smoothing via convolution with the Green operator, the identity admits both first- and second-order scaling limits. The first-order limit is the integral of the Green function on the Sierpiński gasket, while the second-order limit is the distance to the gasket’s corner vertices. An intriguing question is whether the emergence of the distance function in the Sierpiński gasket case is incidental and fails for other graph sequences, or whether a decomposition analogous to Proposition 3.1 also exists more broadly. Identifying such a structure for sandpile identities on more general graphs would allow to analyze scaling limits beyond the Sierpiński gasket, across a wider range of state spaces.

Publisher's Note

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Acknowledgements

The authors are very grateful to the anonymous referee for an exceptionally thorough and constructive review. We specifically acknowledge the referee’s direct mathematical contributions to this revision, including correcting Lemma 4.1, suggesting a cleaner proof of the limit-distance identity, and formulating technical lemmas that substantially strengthened the fractal analysis.

Funding

Open Access funding enabled and organized by Projekt DEAL. The research of E. Sava-Huss and J. Überbacher was funded by Austrian Science Fund (FWF) 10.55776/PAT3123425. For open access purposes, the authors have applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.

Data Availability

There is no external data associated to this manuscript.

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