mjmsMeridian Journal of Mathematical SciencesMeridian J. Math. Sci.2471-35622471-3643Meridian Academic Press10.5555/demo.2022.1029Further evidence: On the structure of the sandpile identity element on Sierpiński gasket graphshttps://orcid.org/0015-0000-4981-051XOlsenErikFunding acquisitionMehtaSanjayProject administrationData curationhttps://orcid.org/0015-0000-5186-9457AchebeThandiweInstitute for Data Science, Kestrel Institute of Technology, CanadaDepartment of Statistics, Cascadia Polytechnic, United StatesSchool of Environmental Sciences, Savanna Institute of Public Health, Kenya
One author is a consultant for a company that manufactures related equipment. The company had no role in the design of the study or the interpretation of the results.
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.
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.
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\begin{document}$$G=(V \cup \{s\}, E)$$\end{document}G=(V∪{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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}^2$$\end{document}Z2 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
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\begin{document}$$\textsf{id}_n$$\end{document}idn be the identity of the sandpile group of the n-th approximation graph \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn of the Sierpiński gasket SG, for \documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}$$\end{document}n∈N. Denote by \documentclass[12pt]{minimal}
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\begin{document}$$g_n$$\end{document}gn the Green function of the simple random walk on \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn stopped at the corners, and let G be the Green function with Dirichlet boundary conditions on the corner vertices on the Sierpiński gasket SG.
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\begin{document}$$g_n*\textsf{id}_n$$\end{document}gn∗idn grows asymptotically as \documentclass[12pt]{minimal}
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\begin{document}$$5^n$$\end{document}5n, and for all x∈ SGx∈SG we have \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{5^n}(g_n*\textsf{id}_n)^{\blacktriangle }(x)=4\int _{SG}G(x,y)d\mu (y). \end{aligned}$$\end{document}limn→∞15n(gn∗idn)▴(x)=4∫SGG(x,y)dμ(y).
The second order term of \documentclass[12pt]{minimal}
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\begin{document}$$g_n*\textsf{id}_n$$\end{document}gn∗idn is given by the graph distance to the corners of \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn\documentclass[12pt]{minimal}
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\begin{document}$$\lim _{n\rightarrow \infty }\frac{1}{2^n}\biggl (g_n*\textsf{id}_n-\frac{8}{3}\sum _{y\in V_{SG_n}}g_n(y,\cdot )\biggr )^{\blacktriangle }(x)=-\frac{1}{3}\textsf{d}(x),$$\end{document}limn→∞12n(gn∗idn-83∑y∈VSGngn(y,·))▴(x)=-13d(x), where \documentclass[12pt]{minimal}
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\begin{document}$$\textsf{d}(x)$$\end{document}d(x) is the shortest path distance from x to the three corners in SG.
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\begin{document}$$x\in V_*$$\end{document}x∈V∗ it holds that \documentclass[12pt]{minimal}
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\begin{document}$$\lim _{n\rightarrow \infty }\frac{1}{2^n}\biggl (g_n*\textsf{id}_n(x)-4\cdot 5^n\int _{SG}G(x,y)d\mu (y)\biggr )=-\frac{1}{3}\textsf{d}(x).$$\end{document}limn→∞12n(gn∗idn(x)-4·5n∫SGG(x,y)dμ(y))=-13d(x).
Before proving Theorem 1.1, we give an informal and intuitive description of the different scaling factors and objects appearing in the statement:
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\begin{document}$$5^n$$\end{document}5n represents the time acceleration factor required for random walks on \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn 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 \documentclass[12pt]{minimal}
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\begin{document}$$\textsf{id}_n$$\end{document}idn as \documentclass[12pt]{minimal}
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\begin{document}$$n \rightarrow \infty $$\end{document}n→∞; see Fig. 3.
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 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 \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn, we obtain \documentclass[12pt]{minimal}
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\begin{document}$$g_n * \textsf{id}_n = \frac{8}{3}\, h_n - \frac{1}{3}\, d_n$$\end{document}gn∗idn=83hn-13dn, where \documentclass[12pt]{minimal}
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\begin{document}$$d_n$$\end{document}dn denotes the distance to the closest corner, and \documentclass[12pt]{minimal}
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\begin{document}$$h_n$$\end{document}hn is the function with Laplacian equal to 1 at every non-corner vertex of \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn 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 \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn, 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 \documentclass[12pt]{minimal}
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\begin{document}$$G = (V \cup \{s\}, E)$$\end{document}G=(V∪{s},E) be a finite, connected, undirected graph with a distinguished vertex s (the sink). For \documentclass[12pt]{minimal}
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\begin{document}$$v \in V \cup \{s\}$$\end{document}v∈V∪{s}, let \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{deg}_G(v)$$\end{document}degG(v) denote its degree in G, i.e., the number of its neighbors; when the graph is clear from context, we write \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{deg}(v)$$\end{document}deg(v). For vertices v, w ∈ Vv,w∈V, we write v ∼ wv∼w if they are adjacent in G (connected by an edge).
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\begin{document}$$\sigma : V \rightarrow \mathbb {N}$$\end{document}σ:V→N that assigns to each vertex v ∈ Vv∈V the number of particles at v. A vertex v is unstable if \documentclass[12pt]{minimal}
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\begin{document}$$\sigma (v) \ge \textrm{deg}(v)$$\end{document}σ(v)≥deg(v); otherwise it is stable. The configuration σ σ is called stable if it is stable at every vertex. For each vertex v ∈ Vv∈V, define a toppling \documentclass[12pt]{minimal}
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\begin{document}$$T_v$$\end{document}Tv at v by removing \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{deg}(v)$$\end{document}deg(v) particles from v and sending one particle along each incident edge. Writing \documentclass[12pt]{minimal}
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\begin{document}$$a_{vw}$$\end{document}avw for the number of edges between v and w, the toppling redistributes particles as follows:\documentclass[12pt]{minimal}
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\begin{document}$$ (T_v \sigma )(u) = {\left\{ \begin{array}{ll} \sigma (u) - \textrm{deg}(v), & u = v, \\ \sigma (u) + a_{vu}, & u \sim v, \\ \sigma (u), & \text {otherwise}. \end{array}\right. } $$\end{document}(Tvσ)(u)=σ(u)-deg(v),u=v,σ(u)+avu,u∼v,σ(u),otherwise.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 σ. Any particles sent to the sink are permanently removed from the system. We view σ 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 \documentclass[12pt]{minimal}
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\begin{document}$$\Delta \in \textrm{Mat}_{|V \cup \{s\}| \times |V \cup \{s\}|}(\mathbb {Z})$$\end{document}Δ∈Mat|V∪{s}|×|V∪{s}|(Z) is the matrix with entries\documentclass[12pt]{minimal}
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\begin{document}$$ \Delta _{vw} = {\left\{ \begin{array}{ll} - a_{vw}, & v \ne w,\ v \sim w, \\ \textrm{deg}(v), & v = w, \\ 0, & \text {otherwise}. \end{array}\right. } $$\end{document}Δvw=-avw,v≠w,v∼w,deg(v),v=w,0,otherwise.Let Δ 'Δ′ denote the reduced Laplacian of G with respect to s, obtained by restricting Δ Δ to V × VV×V (i.e., deleting the row and column corresponding to the sink). Then a toppling at v is given by\documentclass[12pt]{minimal}
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\begin{document}$$ T_v \sigma = \sigma - \Delta ' \delta _v, $$\end{document}Tvσ=σ-Δ′δv,where \documentclass[12pt]{minimal}
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\begin{document}$$\delta _v$$\end{document}δv is the vector with a 1 at index v and 0 elsewhere. This formulation immediately shows that topplings commute (the model is abelian):\documentclass[12pt]{minimal}
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\begin{document}$$ T_v T_w \sigma = T_w T_v \sigma \quad \text {for all } v,w \in V. $$\end{document}TvTwσ=TwTvσfor allv,w∈V.The stabilization \documentclass[12pt]{minimal}
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\begin{document}$$\sigma ^\circ $$\end{document}σ∘ of a sandpile σ σ is the unique stable configuration obtained by performing all legal topplings. This is well defined: there exists a sequence of vertices \documentclass[12pt]{minimal}
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\begin{document}$$v_1,\ldots ,v_n$$\end{document}v1,…,vn—unique up to permutation—such that all topplings are legal and the resulting configuration \documentclass[12pt]{minimal}
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\begin{document}$$\sigma ^\circ = T_{v_n}\cdots T_{v_1}\sigma $$\end{document}σ∘=Tvn⋯Tv1σ 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 \documentclass[12pt]{minimal}
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\begin{document}$$(X_i)_{i \in \mathbb {N}}$$\end{document}(Xi)i∈N be i.i.d. uniformly distributed on V, and start from any stable configuration \documentclass[12pt]{minimal}
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\begin{document}$$\sigma _0$$\end{document}σ0. Define the Markov chain \documentclass[12pt]{minimal}
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\begin{document}$$(\sigma _n)_{n \in \mathbb {N}}$$\end{document}(σn)n∈N by\documentclass[12pt]{minimal}
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\begin{document}$$ \sigma _{n+1} = (\sigma _n + \delta _{X_n})^\circ , $$\end{document}σn+1=(σn+δXn)∘,i.e., at time-step n, given \documentclass[12pt]{minimal}
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\begin{document}$$\sigma _n$$\end{document}σn, we add one particle to a uniformly chosen vertex in G and then we stabilize \documentclass[12pt]{minimal}
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\begin{document}$$\sigma _n + \delta _{X_n}$$\end{document}σn+δXn. The chain \documentclass[12pt]{minimal}
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\begin{document}$$(\sigma _n)_{n \in \mathbb {N}}$$\end{document}(σn)n∈N is called the abelian sandpile Markov chain, and its set of recurrent states forms an abelian group denoted \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}_G$$\end{document}RG, where the group operation \documentclass[12pt]{minimal}
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\begin{document}$$\oplus : \mathcal {R}_G \times \mathcal {R}_G \rightarrow \mathcal {R}_G$$\end{document}⊕:RG×RG→RG is the pointwise addition followed by stabilization \documentclass[12pt]{minimal}
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\begin{document}$$\sigma \oplus \omega = (\sigma + \omega )^\circ $$\end{document}σ⊕ω=(σ+ω)∘. Since \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}_G$$\end{document}RG is an abelian group, it has a neutral element under ⊕ ⊕: a recurrent configuration \documentclass[12pt]{minimal}
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\begin{document}$$\textsf{id}_G$$\end{document}idG, called the sandpile identity of G.
The first three Sierpiński gasket approximation graphs \documentclass[12pt]{minimal}
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\begin{document}$$SG_0$$\end{document}SG0, \documentclass[12pt]{minimal}
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\begin{document}$$SG_2$$\end{document}SG2
The Sierpiński gasket. We define Sierpiński gasket approximation graphs \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn, for \documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}$$\end{document}n∈N as in Barlow and Perkins (1988). For \documentclass[12pt]{minimal}
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\begin{document}$$x \in \mathbb {R}^2$$\end{document}x∈R2 and a graph G=(V,E)G=(V,E) that can be embedded in \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^2$$\end{document}R2, we denote by x+Gx+G the graph\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} x+G = (\{x+v: v \in V\}, \, \{(x+a, x+b): (a,b) \in E\}). \end{aligned}$$\end{document}x+G=({x+v:v∈V},{(x+a,x+b):(a,b)∈E}).For \documentclass[12pt]{minimal}
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\begin{document}$$a\in \mathbb {R}$$\end{document}a∈R, we write aG for the graph\documentclass[12pt]{minimal}
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\begin{document}$$aG{:}{=}(\{av:v\in V\},\{(av,aw):(v,w)\in E\}).$$\end{document}aG:=({av:v∈V},{(av,aw):(v,w)∈E}).Let \documentclass[12pt]{minimal}
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\begin{document}$$u_1=(0,0)$$\end{document}u1=(0,0), \documentclass[12pt]{minimal}
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\begin{document}$$u_2 = \frac{1}{2} (1, \sqrt{3})$$\end{document}u2=12(1,3) and \documentclass[12pt]{minimal}
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\begin{document}$$u_3=(1,0)$$\end{document}u3=(1,0). We define \documentclass[12pt]{minimal}
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\begin{document}$$SG_0=(V_{SG_0}, \, E_{SG_0})$$\end{document}SG0=(VSG0,ESG0) to be the complete graph with vertex and edge set given by\documentclass[12pt]{minimal}
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\begin{document}$$ V_{SG_0}=\{u_1, u_2, u_3\} \quad \text {and} \quad E_{SG_0}=\{\{u_1,u_2\},\{u_2,u_3\},\{u_1,u_3\}\}\,. $$\end{document}VSG0={u1,u2,u3}andESG0={{u1,u2},{u2,u3},{u1,u3}}.Recursively, for n ≥ 1n≥1 the n-th Sierpiński gasket approximation graph \documentclass[12pt]{minimal}
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\begin{document}$$SG_n=(V_{SG_n}, E_{SG_n})$$\end{document}SGn=(VSGn,ESGn) is\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} SG_n = \frac{1}{2} \bigl (SG_{n-1} \cup (u_2 +SG_{n-1}) \cup (u_3 + SG_{n-1})\bigr ). \end{aligned}$$\end{document}SGn=12(SGn-1∪(u2+SGn-1)∪(u3+SGn-1)).Throughout this paper, we will also use the less cumbersome notation \documentclass[12pt]{minimal}
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\begin{document}$$V_n$$\end{document}Vn for the set of vertices in the n-th iteration. Notice that \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn is obtained from \documentclass[12pt]{minimal}
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\begin{document}$$SG_{n-1}$$\end{document}SGn-1 by copying \documentclass[12pt]{minimal}
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\begin{document}$$SG_{n-1}$$\end{document}SGn-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 \documentclass[12pt]{minimal}
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\begin{document}$$SG_0$$\end{document}SG0, \documentclass[12pt]{minimal}
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\begin{document}$$SG_1$$\end{document}SG1 and \documentclass[12pt]{minimal}
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\begin{document}$$SG_2$$\end{document}SG2. Define the union of the vertices of all the approximation graphs\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} V_*= \bigcup _{n \in \mathbb {N}} V_{n}. \end{aligned}$$\end{document}V∗=⋃n∈NVn.The Sierpiński gasket is defined as the closure of \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^2$$\end{document}R2.
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 \documentclass[12pt]{minimal}
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\begin{document}$$i \in \{1,2,3\}$$\end{document}i∈{1,2,3}, define \documentclass[12pt]{minimal}
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\begin{document}$$\psi _i: \mathbb {R}^2 \rightarrow \mathbb {R}^2$$\end{document}ψi:R2→R2 by\documentclass[12pt]{minimal}
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\begin{document}$$u_1, u_2, u_3$$\end{document}u1,u2,u3 are the vertices of the previously defined equilateral triangle. For a word \documentclass[12pt]{minimal}
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\begin{document}$$w=w_1w_2\ldots w_n\in \{1,2,3\}^n$$\end{document}w=w1w2…wn∈{1,2,3}n of length \documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}$$\end{document}n∈N, we write \documentclass[12pt]{minimal}
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\begin{document}$$\psi _w=\psi _{w_n}\circ \cdots \circ \psi _{w_1}$$\end{document}ψw=ψwn∘⋯∘ψw1. We take the σ σ-algebra of measurable sets on SG to be generated by sets of the form \documentclass[12pt]{minimal}
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\begin{document}$$\bigl \{\psi _w(SG): w \in \bigcup _{m \ge 1} \{1,2,3\}^m\bigr \}$$\end{document}{ψw(SG):w∈⋃m≥1{1,2,3}m}. The standard self-similar measure μ μ is the unique measure with \documentclass[12pt]{minimal}
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\begin{document}$$\mu (\psi _w(SG))=3^{-|w|}$$\end{document}μ(ψw(SG))=3-|w|, where |w| is the length of the word w. Notice that this standard self-similar measure is up to scalar multiples a \documentclass[12pt]{minimal}
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\begin{document}$$\log _2(3)$$\end{document}log2(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\documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn. Let \documentclass[12pt]{minimal}
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\begin{document}$$(X_t^n)_{t \in \mathbb {N}}$$\end{document}(Xtn)t∈N denote the simple random walk on \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn, i.e., the Markov chain with transition probabilities given by\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} p(x,y)= {\left\{ \begin{array}{ll} \frac{1}{4}, \quad \text {if } x \underset{n}{\sim }\ y,\, x \ne u_i\\ \frac{1}{2}, \quad \text {if } x \underset{n}{\sim }\ y,\, x = u_i \end{array}\right. } \end{aligned}$$\end{document}p(x,y)=14,ifx∼ny,x≠ui12,ifx∼ny,x=uifor \documentclass[12pt]{minimal}
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\begin{document}$$i \in \{1,2,3\}$$\end{document}i∈{1,2,3}, where \documentclass[12pt]{minimal}
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\begin{document}$$x\underset{n}{\sim }y$$\end{document}x∼ny means that \documentclass[12pt]{minimal}
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\begin{document}$$x,y\in V_{SG_n}$$\end{document}x,y∈VSGn are neighbours in \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn. For \documentclass[12pt]{minimal}
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\begin{document}$$x\in V_{SG_n}$$\end{document}x∈VSGn, denote by \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {E}_x$$\end{document}Ex the expectation with respect to the probability measure \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {P}(\cdot |X^n_0=x)$$\end{document}P(·|X0n=x). The Green function \documentclass[12pt]{minimal}
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\begin{document}$$g_n(x,y)$$\end{document}gn(x,y) on \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn is the expected number of visits to y by the walk started at x before it hits one of the three corner vertices \documentclass[12pt]{minimal}
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\begin{document}$$ g_n(x,y) = \mathbb {E}_x\!\left[ \sum _{t=0}^{\tau _n-1} \textbf{1}_{\{X_t^n = y\}}\right] , \quad \text {where } \tau _n = \inf \left\{ t \ge 0: X_t^n \in \{u_1,u_2,u_3\}\right\} . $$\end{document}gn(x,y)=Ex∑t=0τn-11{Xtn=y},whereτn=inft≥0:Xtn∈{u1,u2,u3}.If we write \documentclass[12pt]{minimal}
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\begin{document}$$\Delta _n$$\end{document}Δn for the graph Laplacian on \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn, then it holds that\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _n g_n(x,\cdot )(y)=\delta _x(y), \quad \text {for all} \quad x,y\in SG_n\backslash \{u_1,u_2,u_3\}.$$\end{document}Δngn(x,·)(y)=δx(y),for allx,y∈SGn\{u1,u2,u3}.Piecewise constant continuation toSG. For x ∈ SGx∈SG, define \documentclass[12pt]{minimal}
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\begin{document}$$x_n \in V_{SG_n}$$\end{document}xn∈VSGn as \documentclass[12pt]{minimal}
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\begin{document}$$x_n = \textrm{argmin}\{d(x,y): y \in SG_n\}$$\end{document}xn=argmin{d(x,y):y∈SGn}. For uniqueness, if there are several minimizers, choose the one furthest to the right. For a function \documentclass[12pt]{minimal}
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\begin{document}$$f:V_{SG_n} \rightarrow \mathbb {R}$$\end{document}f:VSGn→R we define the piecewise constant continuation of f to SG by \documentclass[12pt]{minimal}
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\begin{document}$$f^\blacktriangle : SG \rightarrow \mathbb {R}, f^\blacktriangle (x)= f(x_n)$$\end{document}f▴:SG→R,f▴(x)=f(xn).
Laplacian operator onSG. The Laplacian operator on the Sierpiński gasket SG, denoted by Δ Δ, can be defined via a weak formulation using an energy form on SG. For \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {E}_n$$\end{document}En on \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {E}_n(f,g)=\bigg (\frac{5}{3}\bigg )^n\sum _{x\underset{n}{\sim }\ y}\bigl (f(x)-f(y)\bigr )\bigl (g(x)-g(y)\bigr ),$$\end{document}En(f,g)=(53)n∑x∼ny(f(x)-f(y))(g(x)-g(y)),where \documentclass[12pt]{minimal}
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\begin{document}$$f,g:SG\rightarrow \mathbb {R}$$\end{document}f,g:SG→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)(n+1)-st. For details, see Strichartz (2006, Section 1.3). Using these finite iteration energy forms \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {E}_n$$\end{document}En we define an energy form on SG via\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {E}(f,g)=\lim _{n\rightarrow \infty }\mathcal {E}_n(f|_{V_{SG_n}},g|_{V_{SG_n}}).$$\end{document}E(f,g)=limn→∞En(f|VSGn,g|VSGn).Note that the limit is not necessarily finite. So, we also define the domain of the energy functional as\documentclass[12pt]{minimal}
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\begin{document}$$\operatorname {dom}(\mathcal {E}){:}{=} \{ f: SG \rightarrow \mathbb {R} \mid \mathcal {E}(f,f) < \infty \}.$$\end{document}dom(E):={f:SG→R∣E(f,f)<∞}.With this energy form, we define a weak Laplacian on SG. For \documentclass[12pt]{minimal}
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\begin{document}$$f: SG \rightarrow \mathbb {R}$$\end{document}f:SG→R, we write \documentclass[12pt]{minimal}
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\begin{document}$$u \in \operatorname {dom}(\Delta )$$\end{document}u∈dom(Δ) and Δ u = fΔu=f if and only if\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal {E}(u, v) = - \int _{SG} f\, v \, d\mu \quad \text {for all } v \in \operatorname {dom}(\mathcal {E}). $$\end{document}E(u,v)=-∫SGfvdμfor allv∈dom(E).Green function onSG. For Theorem 1.1, we must specify the function on SG that arises as the scaling limit of the discrete Green functions \documentclass[12pt]{minimal}
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\begin{document}$$g_n$$\end{document}gn 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\documentclass[12pt]{minimal}
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\begin{document}$$ G(x,y){:}{=} \lim _{M \rightarrow \infty } \sum _{m=0}^{M} \sum _{z,z' \in V_{SG_{m+1}} {\setminus } V_{SG_m}} g(z,z')\, \Psi _{z}^{m+1}(x)\, \Psi _{z'}^{m+1}(y), $$\end{document}G(x,y):=limM→∞∑m=0M∑z,z′∈VSGm+1\VSGmg(z,z′)Ψzm+1(x)Ψz′m+1(y),where\documentclass[12pt]{minimal}
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\begin{document}$$ g(z,z') =\! {\left\{ \begin{array}{ll} \dfrac{9}{50}\Big (\dfrac{3}{5}\Big )^m, & z = z' \ \text {and}\ z \in V_{SG_{m+1}} {\setminus } V_{SG_m},\\ \dfrac{3}{50}\Big (\dfrac{3}{5}\Big )^m, & z \ne z',\ z,z' \in V_{SG_{m+1}} {\setminus } V_{SG_m},\ \text {and}\ z,z' \!\in \! \psi _w(SG)\ \text {for some } |w|=m, \end{array}\right. } $$\end{document}g(z,z′)=950(35)m,z=z′andz∈VSGm+1\VSGm,350(35)m,z≠z′,z,z′∈VSGm+1\VSGm,andz,z′∈ψw(SG)for some|w|=m,and, for \documentclass[12pt]{minimal}
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\begin{document}$$z \in V_{SG_m}$$\end{document}z∈VSGm, the level-m harmonic spline \documentclass[12pt]{minimal}
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\begin{document}$$\Psi _z^m$$\end{document}Ψzm is the unique piecewise harmonic function on SG satisfying\documentclass[12pt]{minimal}
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\begin{document}$$ \Psi _z^m(x) = {\left\{ \begin{array}{ll} 1, & x = z,\\ 0, & x \in V_{m} {\setminus } \{z\}. \end{array}\right. } $$\end{document}Ψzm(x)=1,x=z,0,x∈Vm\{z}.A key feature of the Green function on SG is that, up to the scaling factor \documentclass[12pt]{minimal}
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\begin{document}$$(3/5)^n$$\end{document}(3/5)n, it matches the stopped Green function \documentclass[12pt]{minimal}
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\begin{document}$$g_n$$\end{document}gn on the vertices of \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn.
Lemma 2.1
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\begin{document}$$n \in \mathbb {N}$$\end{document}n∈N and all \documentclass[12pt]{minimal}
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\begin{document}$$x,y \in V_{n}$$\end{document}x,y∈Vn it holds \documentclass[12pt]{minimal}
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\begin{document}$$\left( \frac{3}{5}\right) ^n g_n(x,y) = G(x,y).$$\end{document}35ngn(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
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\begin{document}$$h:SG\rightarrow \mathbb {R}$$\end{document}h:SG→R be a harmonic function on \documentclass[12pt]{minimal}
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\begin{document}$$SG\backslash V_0$$\end{document}SG\V0. Then for every \documentclass[12pt]{minimal}
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\begin{document}$$m,n\in \mathbb {N}$$\end{document}m,n∈N with m>nm>n and \documentclass[12pt]{minimal}
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\begin{document}$$x\in V_n\backslash V_0$$\end{document}x∈Vn\V0 it holds that\documentclass[12pt]{minimal}
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\begin{document}$$\frac{1}{\mu (B(x,2^{-m}))}\int _{B(x,2^{-m})}h(y)d\mu (y)=h(x).$$\end{document}1μ(B(x,2-m))∫B(x,2-m)h(y)dμ(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 \documentclass[12pt]{minimal}
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\begin{document}$$x_{1},x_{2},x_{3} \in V_{n}$$\end{document}x1,x2,x3∈Vn are the corner vertices and\documentclass[12pt]{minimal}
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\begin{document}$$y_{1},y_{2},y_{3}\in V_{n+1}\backslash V_{n}$$\end{document}y1,y2,y3∈Vn+1\Vn are the corresponding cut points (each \documentclass[12pt]{minimal}
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\begin{document}$$y_i$$\end{document}yi opposite \documentclass[12pt]{minimal}
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\begin{document}$$x_i$$\end{document}xi), the rule gives \documentclass[12pt]{minimal}
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\begin{document}$$h=h(x_{1})\psi _{1}+h(x_{2})\psi _{2}+h(x_{3})\psi _{3}$$\end{document}h=h(x1)ψ1+h(x2)ψ2+h(x3)ψ3 on this n-cell, where \documentclass[12pt]{minimal}
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\begin{document}$$\psi _{i}$$\end{document}ψi is the unique harmonic spline with \documentclass[12pt]{minimal}
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\begin{document}$$\psi _{i}(x_{j})=\delta _{ij}$$\end{document}ψi(xj)=δij, and the algorithm gives\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \psi _{1}(y_{1})=\frac{1}{5},\quad \psi _{1}(y_{2})=\psi _{1}(y_{3})=\frac{2}{5}, \end{aligned}$$\end{document}ψ1(y1)=15,ψ1(y2)=ψ1(y3)=25,The other values of \documentclass[12pt]{minimal}
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\begin{document}$$\psi _{i}(y_{j})$$\end{document}ψi(yj) are found via dihedral symmetry. Consequently h is discrete harmonic:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} h(y_{1})=\frac{1}{4}(h(x_{2})+h(x_{3})+h(y_{2})+h(y_{3})). \end{aligned}$$\end{document}h(y1)=14(h(x2)+h(x3)+h(y2)+h(y3)).Moreover, since \documentclass[12pt]{minimal}
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\begin{document}$$\sum _{i=1}^{3}\psi _{i}\equiv 1$$\end{document}∑i=13ψi≡1 and \documentclass[12pt]{minimal}
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\begin{document}$$\mu (m\text {-cell})=3^{-m}$$\end{document}μ(m-cell)=3-m for every m≥ nm≥n we deduce by symmetry that for every (n+1)(n+1)-subcell \documentclass[12pt]{minimal}
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\begin{document}$$\int _{\mathcal {T}_{n+1}}\psi _{i}d\mu =3^{-(n+2)}$$\end{document}∫Tn+1ψidμ=3-(n+2) for \documentclass[12pt]{minimal}
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\begin{document}$$i\in \{1,2,3\}$$\end{document}i∈{1,2,3}. We now establish the mean-value property of h over the ball \documentclass[12pt]{minimal}
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\begin{document}$$B(y_{1},2^{-(n+1)})$$\end{document}B(y1,2-(n+1)):\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{1}{\mu (B(y_{1},2^{-(n+1)}))}\int _{B(y_{1},2^{-(n+1)})}h~d\mu =h(y_{1}). \end{aligned}$$\end{document}1μ(B(y1,2-(n+1)))∫B(y1,2-(n+1))hdμ=h(y1).Using (1) and the above integrals of \documentclass[12pt]{minimal}
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\begin{document}$$\psi _{i}$$\end{document}ψi, we get\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _{B(y_{1},2^{-(n+1)})}h~d\mu&=\big (h(y_{1})+h(x_{3})+h(y_{2})\big )\cdot 3^{-(n+2)}+\big (h(y_{1})+h(x_{2})+h(y_{3})\big )\cdot 3^{-(n+2)} \\&=\frac{1}{3}\cdot 3^{-(n+1)}\cdot \big (h(x_{2})+h(x_{3})+h(y_{2})+h(y_{3})+2h(y_{1})\big ) \\&=\frac{1}{3}\cdot 3^{-(n+1)}\cdot 6h(y_{1}), \end{aligned}$$\end{document}∫B(y1,2-(n+1))hdμ=(h(y1)+h(x3)+h(y2))·3-(n+2)+(h(y1)+h(x2)+h(y3))·3-(n+2)=13·3-(n+1)·(h(x2)+h(x3)+h(y2)+h(y3)+2h(y1))=13·3-(n+1)·6h(y1),where we used (2) in the last equation. Dividing the last expression by \documentclass[12pt]{minimal}
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\begin{document}$$\mu (B(y_{1},2^{-(n+1)}))=2\cdot 3^{-(n+1)}$$\end{document}μ(B(y1,2-(n+1)))=2·3-(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
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\begin{document}$$R_{eff}(A,B)$$\end{document}Reff(A,B) be the effective resistance between subsets A, B⊂ SGA,B⊂SG, subject to Dirichlet boundary conditions on \documentclass[12pt]{minimal}
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\begin{document}$$V_{0}$$\end{document}V0. Let G(x, y) be the Green’s function on SG subject to Dirichlet boundary conditions on \documentclass[12pt]{minimal}
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\begin{document}$$V_{0}$$\end{document}V0. Then the following holds:
For every x∈ SGx∈SG, \documentclass[12pt]{minimal}
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\begin{document}$$G(x,x)=R_{eff}(x,V_{0})$$\end{document}G(x,x)=Reff(x,V0).
For every x,y∈ SGx,y∈SG, \documentclass[12pt]{minimal}
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\begin{document}$$G(x,y)=\frac{1}{2}(G(x,x)+G(y,y)-R_{eff}(x,y))$$\end{document}G(x,y)=12(G(x,x)+G(y,y)-Reff(x,y)).
For every x,y∈ SGx,y∈SG we have G(x,x)≥ G(x,y)G(x,x)≥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∈ SGx,y,z∈SG we have\documentclass[12pt]{minimal}
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\begin{document}$$|G(x,y)-G(x,z)|\le R_{eff}(y,z),$$\end{document}|G(x,y)-G(x,z)|≤Reff(y,z),and consequently\documentclass[12pt]{minimal}
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\begin{document}$$0\le G(x,x)-G(x,y)\le R_{eff}(x,y).$$\end{document}0≤G(x,x)-G(x,y)≤Reff(x,y).
Proof
Using (1) and (2) in Proposition 2.1, we obtain\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} G(x,y)-G(x,z)&=\frac{1}{2}[G(y,y)-G(z,z)+R_{eff}(x,z)-R_{eff}(x,y)] \\&=\frac{1}{2}[R_{eff}(y,V_{0})-R_{eff}(z,V_{0})+R_{eff}(x,z)-R_{eff}(x,y)]. \end{aligned}$$\end{document}G(x,y)-G(x,z)=12[G(y,y)-G(z,z)+Reff(x,z)-Reff(x,y)]=12[Reff(y,V0)-Reff(z,V0)+Reff(x,z)-Reff(x,y)].Note that \documentclass[12pt]{minimal}
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\begin{document}$$V_{0}$$\end{document}V0 is grounded and hence can be collapsed to a single vertex. Meanwhile, the resistance metric satisfies the triangle inequality, so\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |G(x,y)-G(x,z)|&\le \frac{1}{2}(|R_{eff}(y,V_{0})-R_{eff}(z,V_{0})|+|R_{eff}(x,z)-R_{eff}(x,y)|) \\&\le \frac{1}{2}(R_{eff}(y,z)+R_{eff}(y,z))=R_{eff}(y,z), \end{aligned}$$\end{document}|G(x,y)-G(x,z)|≤12(|Reff(y,V0)-Reff(z,V0)|+|Reff(x,z)-Reff(x,y)|)≤12(Reff(y,z)+Reff(y,z))=Reff(y,z),proving the first inequality. This together with (3) from Proposition 3 implies the second part. □ □
Lemma 2.4
There exists a constant C>0C>0 such that the following hold.
For every x,y∈ SGx,y∈SG belonging to the same n-cell, we have \documentclass[12pt]{minimal}
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\begin{document}$$0\le G(x,x)-G(x,y)\le C\left( \frac{3}{5}\right) ^{n},$$\end{document}0≤G(x,x)-G(x,y)≤C35n,
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\begin{document}$$n\in \mathbb {N}$$\end{document}n∈N and y∈ SGy∈SG with \documentclass[12pt]{minimal}
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\begin{document}$$d(y,V_{0})\le 2^{-n}$$\end{document}d(y,V0)≤2-n we have \documentclass[12pt]{minimal}
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\begin{document}$$0\le G(x,y)\le C\left( \frac{3}{5}\right) ^{n}.$$\end{document}0≤G(x,y)≤C35n.
Proof
For the first claim, using Lemma 2.3 it suffices to show that there exists C>0C>0 such that for every x,y∈ SGx,y∈SG belonging to the same n-cell, we have \documentclass[12pt]{minimal}
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\begin{document}$$R_{eff}(x,y)\le C\left( \frac{3}{5}\right) ^{n}$$\end{document}Reff(x,y)≤C35n. 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 \documentclass[12pt]{minimal}
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\begin{document}$$G(x,y)\le G(y,y)=R_{eff}(y,V_{0})$$\end{document}G(x,y)≤G(y,y)=Reff(y,V0). Since y shares the same n-cell with one of the corner vertices in \documentclass[12pt]{minimal}
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\begin{document}$$V_{0}$$\end{document}V0, applying the aforementioned resistance upper bound yields \documentclass[12pt]{minimal}
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\begin{document}$$R_{eff}(y,V_{0})\le C\left( \frac{3}{5}\right) ^{n}$$\end{document}Reff(y,V0)≤C35n. □ □
Sandpiles on the Sierpiński gasket graphs
Boundary conditions on\documentclass[12pt]{minimal}
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\begin{document}$$V_{SG_n}$$\end{document}VSGn has degree 4 (see Fig. 22). Throughout, “sandpile group on \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn” refers to the set of recurrent configurations of the sandpile Markov chain with this boundary, denoted \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}_n$$\end{document}Rn, and we write \documentclass[12pt]{minimal}
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\begin{document}$$\textsf{id}_n$$\end{document}idn for its identity element.
The Sierpiński gasket graphs \documentclass[12pt]{minimal}
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\begin{document}$$SG_0$$\end{document}SG0, \documentclass[12pt]{minimal}
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\begin{document}$$SG_1$$\end{document}SG1 and \documentclass[12pt]{minimal}
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\begin{document}$$SG_2$$\end{document}SG2 with normal boundary conditions
In Kaiser et al. (2024), the authors prove in Theorem 3.2 that for any n ≥ 1n≥1, the identity element of \documentclass[12pt]{minimal}
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\begin{document}$$SG_{n+1}$$\end{document}SGn+1 with normal boundary conditions is given by where \documentclass[12pt]{minimal}
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\begin{document}$$M_n^+$$\end{document}Mn+ (respectively \documentclass[12pt]{minimal}
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\begin{document}$$M_n^-$$\end{document}Mn-) denotes the sandpile configuration obtained from \documentclass[12pt]{minimal}
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\begin{document}$$M_n$$\end{document}Mn by rotating \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn counterclockwise (respectively clockwise) by \documentclass[12pt]{minimal}
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\begin{document}$$120^\circ $$\end{document}120∘. The sandpile identity elements on the Sierpiński gaskets \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn for \documentclass[12pt]{minimal}
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\begin{document}$$n \in \{2,3,4,5\}$$\end{document}n∈{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 \documentclass[12pt]{minimal}
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\begin{document}$$n \in \mathbb {N}$$\end{document}n∈N, define \documentclass[12pt]{minimal}
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\begin{document}$$I_n: V_{SG_n} \rightarrow \mathbb {R}$$\end{document}In:VSGn→R by\documentclass[12pt]{minimal}
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\begin{document}$$I_n(x) = (g_n * \textsf{id}_n)(x){:}{=} \sum _{y \in V_{SG_n}} g_n(x,y)\,\textsf{id}_n(y).$$\end{document}In(x)=(gn∗idn)(x):=∑y∈VSGngn(x,y)idn(y).We have \documentclass[12pt]{minimal}
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\begin{document}$$\Delta _n I_n = \textsf{id}_n$$\end{document}ΔnIn=idn on \documentclass[12pt]{minimal}
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\begin{document}$$V_{SG_n}{\setminus }\{u_1,u_2,u_3\}$$\end{document}VSGn\{u1,u2,u3} and \documentclass[12pt]{minimal}
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\begin{document}$$I_n(u_i)=0$$\end{document}In(ui)=0 for each \documentclass[12pt]{minimal}
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\begin{document}$$i\in \{1,2,3\}$$\end{document}i∈{1,2,3}; that is, \documentclass[12pt]{minimal}
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\begin{document}$$I_n$$\end{document}In vanishes at the corner vertices of \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn. We also define \documentclass[12pt]{minimal}
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\begin{document}$$h_n: V_{SG_n}\rightarrow \mathbb {R}$$\end{document}hn:VSGn→R by\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} h_n(x) = \sum _{y\in V_{SG_n}} g_n(x,y). \end{aligned}$$\end{document}hn(x)=∑y∈VSGngn(x,y).It follows that \documentclass[12pt]{minimal}
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\begin{document}$$\Delta _n h_n(x)=1$$\end{document}Δnhn(x)=1 for all \documentclass[12pt]{minimal}
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\begin{document}$$x\in V_{SG_n}{\setminus }\{u_1,u_2,u_3\}$$\end{document}x∈VSGn\{u1,u2,u3}, and \documentclass[12pt]{minimal}
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\begin{document}$$h_n$$\end{document}hn vanishes at the corner vertices of \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn. Next, define the distance to the corners by\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d_n: V_{SG_n}\rightarrow \mathbb {R},\quad d_n(x)= \min \bigl \{\,d_{SG_n}(x,u_i): i\in \{1,2,3\}\,\bigr \}, \end{aligned}$$\end{document}dn:VSGn→R,dn(x)=min{dSGn(x,ui):i∈{1,2,3}},where \documentclass[12pt]{minimal}
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\begin{document}$$d_{SG_n}$$\end{document}dSGn denotes the graph distance in \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn. In the next proposition, we show that \documentclass[12pt]{minimal}
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\begin{document}$$I_n$$\end{document}In can be expressed as a linear combination of the functions \documentclass[12pt]{minimal}
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\begin{document}$$d_n$$\end{document}dn and \documentclass[12pt]{minimal}
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\begin{document}$$h_n$$\end{document}hn.
The identity element \documentclass[12pt]{minimal}
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\begin{document}$$\textsf{id}_n$$\end{document}idn on \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn for \documentclass[12pt]{minimal}
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\begin{document}$$n \in \{2,3,4,5\}$$\end{document}n∈{2,3,4,5}. Blue dots are vertices with 3 chips and red dots are vertices with 2 chips (color figure online)
Proposition 3.1
For n ≥ 2n≥2 and \documentclass[12pt]{minimal}
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\begin{document}$$I_n=g_n*\textsf{id}_n$$\end{document}In=gn∗idn, it holds that\documentclass[12pt]{minimal}
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\begin{document}$$I_n=-\frac{1}{3}d_n+\frac{8}{3}h_n.$$\end{document}In=-13dn+83hn.
Proof
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\begin{document}$$F_n{:}{=}-\frac{1}{3}d_n+\frac{8}{3}h_n$$\end{document}Fn:=-13dn+83hn and we first show that\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _n F_n(x)=\textsf{id}_n(x) \quad \text {for all} \quad x\in V_{SG_n}\backslash \{u_1,u_2,u_3\}.$$\end{document}ΔnFn(x)=idn(x)for allx∈VSGn\{u1,u2,u3}.Since the functions \documentclass[12pt]{minimal}
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\begin{document}$$d_n$$\end{document}dn and \documentclass[12pt]{minimal}
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\begin{document}$$h_n$$\end{document}hn are rotationally symmetric, it suffices to consider \documentclass[12pt]{minimal}
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\begin{document}$$\Delta _n F_n$$\end{document}ΔnFn only on the top triangle \documentclass[12pt]{minimal}
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\begin{document}$$\bigtriangleup _n^2$$\end{document}△n2. 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 \documentclass[12pt]{minimal}
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\begin{document}$$\triangle _n^2$$\end{document}▵n2 for the vertices of \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn in the upper triangle \documentclass[12pt]{minimal}
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\begin{document}$$\triangle _n^2=\psi _2(V_{SG_{n-1}}\backslash \{u_1,u_2,u_3\})$$\end{document}▵n2=ψ2(VSGn-1\{u1,u2,u3}) and similarly\documentclass[12pt]{minimal}
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\begin{document}$$\triangle _n^1=\psi _1\bigl (V_{SG_{n-1}}\backslash \{u_1,u_2,u_3\}\bigr ),\quad \triangle _n^3=\psi _3\bigl (V_{SG_{n-1}}\backslash \{u_1,u_2,u_3\}\bigr ).$$\end{document}▵n1=ψ1(VSGn-1\{u1,u2,u3}),▵n3=ψ3(VSGn-1\{u1,u2,u3}).
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\begin{document}$$p_1,p_2,p_3$$\end{document}p1,p2,p3 and the subtriangles \documentclass[12pt]{minimal}
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\begin{document}$$\triangle _n^i$$\end{document}▵ni for \documentclass[12pt]{minimal}
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\begin{document}$$i\in \{1,2,3\}$$\end{document}i∈{1,2,3} in \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn
We write \documentclass[12pt]{minimal}
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\begin{document}$$p_1,p_2$$\end{document}p1,p2 and \documentclass[12pt]{minimal}
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\begin{document}$$p_3$$\end{document}p3 for the three cutpoints, so \documentclass[12pt]{minimal}
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\begin{document}$$p_1=\frac{1}{2}u_2$$\end{document}p1=12u2, \documentclass[12pt]{minimal}
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\begin{document}$$p_2=\frac{1}{2}u_3$$\end{document}p2=12u3, \documentclass[12pt]{minimal}
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\begin{document}$$p_3=p_1+p_2$$\end{document}p3=p1+p2; 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=2n=2.
The graph distance from the top vertex \documentclass[12pt]{minimal}
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\begin{document}$$u_2$$\end{document}u2 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 \documentclass[12pt]{minimal}
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\begin{document}$$\bigtriangleup _2^2$$\end{document}△22, we have colored the top vertex \documentclass[12pt]{minimal}
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\begin{document}$$u_2$$\end{document}u2 black. Since the Laplacian of \documentclass[12pt]{minimal}
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\begin{document}$$h_2$$\end{document}h2 is 1 in all non-corner vertices, the Laplacian of \documentclass[12pt]{minimal}
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\begin{document}$$F_2$$\end{document}F2 on \documentclass[12pt]{minimal}
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\begin{document}$$\bigtriangleup _2^2$$\end{document}△22 can be computed as
For the values at the cutpoints we have
This proves the statement for n=2n=2.
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\begin{document}$$\Delta F_n \vert _{SG_n\backslash \{u_1,u_2,u_3\}}=\textsf{id}_n \vert _{SG_n\backslash \{u_1,u_2,u_3\}}$$\end{document}ΔFn|SGn\{u1,u2,u3}=idn|SGn\{u1,u2,u3} holds \documentclass[12pt]{minimal}
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\begin{document}$$n \in \mathbb {N}$$\end{document}n∈N, and we show that the statement holds for n+1n+1. We restrict again our attention to the upper subtriangle \documentclass[12pt]{minimal}
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\begin{document}$$\triangle _{n+1}^2$$\end{document}▵n+12. Within \documentclass[12pt]{minimal}
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\begin{document}$$\triangle _{n+1}^2$$\end{document}▵n+12, the distance function \documentclass[12pt]{minimal}
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\begin{document}$$d_{n+1}$$\end{document}dn+1 coincides with the graph distance to the top corner vertex \documentclass[12pt]{minimal}
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\begin{document}$$u_2$$\end{document}u2. Thus, on \documentclass[12pt]{minimal}
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\begin{document}$$\triangle _{n+1}^2$$\end{document}▵n+12, the distance from the top corner can be depicted as in Fig. 5.
The values of the graph distance function in the subtriangle \documentclass[12pt]{minimal}
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\begin{document}$$\bigtriangleup _{n+1}^2$$\end{document}△n+12
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 \documentclass[12pt]{minimal}
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\begin{document}$$2^{n-1}$$\end{document}2n-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 \documentclass[12pt]{minimal}
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\begin{document}$$\Delta F_{n+1}(x)=\textsf{id}_{n+1}(x)$$\end{document}ΔFn+1(x)=idn+1(x). Now we examine the values of the Laplacian at the cutpoints \documentclass[12pt]{minimal}
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\begin{document}$$c_2',c_2''$$\end{document}c2′,c2′′ and m in \documentclass[12pt]{minimal}
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\begin{document}$$\triangle _{n+1}^2$$\end{document}▵n+12 as shown in Fig. 5. For \documentclass[12pt]{minimal}
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\begin{document}$$c_2'$$\end{document}c2′ we have\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Delta d_{SG_{n+1}}(u_2,c_2')&=\sum _{y \sim c_2'}d_{SG_{n+1}}(u_2,c_2') - d_{SG_{n+1}}(u_2,y)\\&=4 \cdot 2^{n-1} - ((2^{n-1} - 1) + 2^{n-1} + 2 \cdot (2^{n-1}+1)) = -1. \end{aligned}$$\end{document}ΔdSGn+1(u2,c2′)=∑y∼c2′dSGn+1(u2,c2′)-dSGn+1(u2,y)=4·2n-1-((2n-1-1)+2n-1+2·(2n-1+1))=-1.By symmetry, for \documentclass[12pt]{minimal}
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\begin{document}$$c_2''$$\end{document}c2′′ it holds that \documentclass[12pt]{minimal}
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\begin{document}$$\Delta d_{SG_{n+1}}(u_2,c_2'')=-1$$\end{document}ΔdSGn+1(u2,c2′′)=-1, and for m similar computations lead to\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Delta d_{SG_{n+1}}(u_2,m)=4 \cdot 2^n - (2 \cdot (2^n - 1) + 2 \cdot 2^n) =2. \end{aligned}$$\end{document}ΔdSGn+1(u2,m)=4·2n-(2·(2n-1)+2·2n)=2.Hence, on \documentclass[12pt]{minimal}
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\begin{document}$$\triangle _{n+1}^2$$\end{document}▵n+12 we have \documentclass[12pt]{minimal}
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\begin{document}$$\Delta F_{n+1}|_{\triangle _{n+1}^2}=\textsf{id}_{n+1}|_{\triangle _{n+1}^2}$$\end{document}ΔFn+1|▵n+12=idn+1|▵n+12, and by rotational symmetry, the same holds on \documentclass[12pt]{minimal}
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\begin{document}$$\triangle _{n+1}^1$$\end{document}▵n+11 and \documentclass[12pt]{minimal}
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\begin{document}$$\triangle _{n+1}^3$$\end{document}▵n+13. It remains to address the cutpoints \documentclass[12pt]{minimal}
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\begin{document}$$p_1, p_2, p_3$$\end{document}p1,p2,p3 of \documentclass[12pt]{minimal}
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\begin{document}$$SG_{n+1}$$\end{document}SGn+1; see Fig. 4 for the cutpoints of \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn. By symmetry, the values of \documentclass[12pt]{minimal}
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\begin{document}$$\Delta d_{n+1}(\cdot )$$\end{document}Δdn+1(·) at these vertices agree and are equal to\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Delta d_{n+1} (p_1)&= \sum _{v \sim p_1} d_{n+1}(p_1) - d_{n+1}(v) = 4 \cdot 2^n - (2 \cdot 2^n + 2 \cdot (2^n -1)) = 2. \end{aligned}$$\end{document}Δdn+1(p1)=∑v∼p1dn+1(p1)-dn+1(v)=4·2n-(2·2n+2·(2n-1))=2.This completes the argument, yielding \documentclass[12pt]{minimal}
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\begin{document}$$\Delta _n F_n(x) = \textsf{id}_n(x)$$\end{document}ΔnFn(x)=idn(x) for all \documentclass[12pt]{minimal}
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\begin{document}$$n \in \mathbb {N}$$\end{document}n∈N and all \documentclass[12pt]{minimal}
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\begin{document}$$x \in V_{SG_n}{\setminus }\{u_1,u_2,u_3\}$$\end{document}x∈VSGn\{u1,u2,u3}. Define \documentclass[12pt]{minimal}
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\begin{document}$$T_n = F_n - I_n$$\end{document}Tn=Fn-In. Then \documentclass[12pt]{minimal}
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\begin{document}$$\Delta _n T_n(x) = 0$$\end{document}ΔnTn(x)=0 for all \documentclass[12pt]{minimal}
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\begin{document}$$x \in V_{SG_n}{\setminus }\{u_1,u_2,u_3\}$$\end{document}x∈VSGn\{u1,u2,u3} and \documentclass[12pt]{minimal}
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\begin{document}$$T_n(u_i)=0$$\end{document}Tn(ui)=0 for each \documentclass[12pt]{minimal}
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\begin{document}$$i\in \{1,2,3\}$$\end{document}i∈{1,2,3}. By the maximum (–minimum) principle for harmonic functions, \documentclass[12pt]{minimal}
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\begin{document}$$T_n(x)=0$$\end{document}Tn(x)=0 for all \documentclass[12pt]{minimal}
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\begin{document}$$x\in V_{SG_n}$$\end{document}x∈VSGn, hence \documentclass[12pt]{minimal}
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\begin{document}$$I_n = F_n$$\end{document}In=Fn, completing the proof. □ □
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 \documentclass[12pt]{minimal}
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\begin{document}$$d_{SG}(x,y)$$\end{document}dSG(x,y) denote the path distance between points x,y∈ SGx,y∈SG, which can be obtained from the graph distance on \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn via\documentclass[12pt]{minimal}
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\begin{document}$$d_{SG}(x,y)=\lim _{n\rightarrow \infty }\frac{1}{2^n}d_{SG_n}(x_n,y_n),$$\end{document}dSG(x,y)=limn→∞12ndSGn(xn,yn),where \documentclass[12pt]{minimal}
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\begin{document}$$x_n$$\end{document}xn and \documentclass[12pt]{minimal}
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\begin{document}$$y_n$$\end{document}yn are the vertices in \documentclass[12pt]{minimal}
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\begin{document}$$SG_n$$\end{document}SGn closest to x and y respectively. Write\documentclass[12pt]{minimal}
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\begin{document}$$\textsf{d}(x)=\min \{\,d_{SG}(x,u_1),\, d_{SG}(x,u_2),\, d_{SG}(x,u_3)\,\}.$$\end{document}d(x)=min{dSG(x,u1),dSG(x,u2),dSG(x,u3)}.
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\begin{document}$$g_n*\textsf{id}_n(x)-\frac{8}{3}h_n(x)=-\frac{1}{3}d_n(x),$$\end{document}gn∗idn(x)-83hn(x)=-13dn(x),for all \documentclass[12pt]{minimal}
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\begin{document}$$x \in V_{SG_n}$$\end{document}x∈VSGn, from which the claim follows by using\documentclass[12pt]{minimal}
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\begin{document}$$\lim _{n\rightarrow \infty }\frac{1}{2^n}\left( g_n*\textsf{id}_n-\frac{8}{3}h_n\right) ^\blacktriangle (x)=\lim _{n\rightarrow \infty }-\frac{1}{3\cdot 2^n}d_n^\blacktriangle (x)=-\frac{1}{3}\textsf{d}(x).$$\end{document}limn→∞12ngn∗idn-83hn▴(x)=limn→∞-13·2ndn▴(x)=-13d(x).□ □
We now address statement (I3) of Theorem 1.1. The first step is to establish that the functions \documentclass[12pt]{minimal}
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\begin{document}$$h_n$$\end{document}hn provide a good approximation to the integral of the Green function G over the Sierpiński gasket.
Lemma 4.1
There exists a constant \documentclass[12pt]{minimal}
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\begin{document}$$c_{*}>0$$\end{document}c∗>0 such that for all \documentclass[12pt]{minimal}
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\begin{document}$$x\in V_{*}$$\end{document}x∈V∗, it holds that\documentclass[12pt]{minimal}
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\begin{document}$$ \limsup _{n\rightarrow \infty }\left| 3\cdot 5^{n}\int _{SG}G(x,y)d\mu (y)-2\sum _{y\in V_{n}}g_{n}(x,y)\right| \le c_{*}. $$\end{document}lim supn→∞3·5n∫SGG(x,y)dμ(y)-2∑y∈Vngn(x,y)≤c∗.
Proof
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\begin{document}$$x\in V_{0}$$\end{document}x∈V0, then G(x,y)=0G(x,y)=0 for all y∈ SGy∈SG, and \documentclass[12pt]{minimal}
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\begin{document}$$g_{n}(x,y)=0$$\end{document}gn(x,y)=0 for all \documentclass[12pt]{minimal}
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\begin{document}$$y\in V_{n}$$\end{document}y∈Vn, making the lemma vacuously true.
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\begin{document}$$x\in V_{*}\backslash V_{0}$$\end{document}x∈V∗\V0. By definition, \documentclass[12pt]{minimal}
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\begin{document}$$x\in V_{n}$$\end{document}x∈Vn for all sufficiently large \documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}$$\end{document}n∈N. Also, given \documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}$$\end{document}n∈N we may partition SG into balls of diameter \documentclass[12pt]{minimal}
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\begin{document}$$2^{-(n+1)}$$\end{document}2-(n+1) in the Euclidean distance centered at each \documentclass[12pt]{minimal}
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\begin{document}$$z\in V_{n}$$\end{document}z∈Vn, \documentclass[12pt]{minimal}
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\begin{document}$$SG=\bigcup _{z\in V_{n}}B(z,2^{-(n+1)})$$\end{document}SG=⋃z∈VnB(z,2-(n+1)). If \documentclass[12pt]{minimal}
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\begin{document}$$z\in V_{n}\backslash V_{0}$$\end{document}z∈Vn\V0, then the closed ball \documentclass[12pt]{minimal}
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\begin{document}$$B(z,2^{-(n+1)})$$\end{document}B(z,2-(n+1)) consists of two (n+1)(n+1)-cells adjoined at z, and if \documentclass[12pt]{minimal}
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\begin{document}$$z\in V_{0}$$\end{document}z∈V0, then \documentclass[12pt]{minimal}
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\begin{document}$$B(z,2^{-(n+1)})$$\end{document}B(z,2-(n+1)) is the single (n+1)(n+1)-cell containing the corner vertex z. As a result, under the standard self-similar probability measure μ μ, the mass assigned to each ball is\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mu \big (B(z,2^{-(n+1)})\big ) = {\left\{ \begin{array}{ll} 2\cdot 3^{-(n+1)}, & \text {if } z\in V_{n}\backslash V_{0} \\ 3^{-(n+1)}, & \text {if } z\in V_{0} \end{array}\right. } \end{aligned}$$\end{document}μ(B(z,2-(n+1)))=2·3-(n+1),ifz∈Vn\V03-(n+1),ifz∈V0Note that while two different balls may intersect, the set of their intersection has μ 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\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _{SG}G(x,y)d\mu (y)&=\sum _{z\in V_{n}}\int _{B(z,2^{-(n+1)})}G(x,y)d\mu (y) \\&=\sum _{z\in V_{n}\backslash V_{0}}\int _{B(z,2^{-(n+1)})}G(x,y)d\mu (y)+\sum _{z\in V_{0}}\int _{B(z,2^{-(n+1)})}G(x,y)d\mu (y) \\&{=}{:}I_{bulk}+I_{boundary}. \end{aligned}$$\end{document}∫SGG(x,y)dμ(y)=∑z∈Vn∫B(z,2-(n+1))G(x,y)dμ(y)=∑z∈Vn\V0∫B(z,2-(n+1))G(x,y)dμ(y)+∑z∈V0∫B(z,2-(n+1))G(x,y)dμ(y)=:Ibulk+Iboundary.To analyze \documentclass[12pt]{minimal}
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\begin{document}$$I_{bulk}$$\end{document}Ibulk, by the harmonic spline construction of the Green’s function, the function G(x,· )G(x,·) is harmonic on \documentclass[12pt]{minimal}
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\begin{document}$$B(z,2^{-(n+1)})$$\end{document}B(z,2-(n+1)) as long as z≠ xz≠x and \documentclass[12pt]{minimal}
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\begin{document}$$z\notin V_{0}$$\end{document}z∉V0. Applying Lemma 2.2, we get\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{1}{\mu (B(z,2^{-(n+1)}))}\int _{B(z,2^{-(n+1)})}G(x,y)d\mu (y)=G(x,z) \quad \text {for every } z\in V_{n}\backslash V_{0},~z\ne x. \end{aligned}$$\end{document}1μ(B(z,2-(n+1)))∫B(z,2-(n+1))G(x,y)dμ(y)=G(x,z)for everyz∈Vn\V0,z≠x.which together with Eqs. (6) and (7) yields\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} I_{bulk}&=2\cdot 3^{-(n+1)}\sum _{z\in V_{n}\backslash V_{0},z\ne x}G(x,z)+\int _{B(x,2^{-(n+1)})}G(x,y)d\mu (y) \\&=2\cdot 3^{-(n+1)}\sum _{z\in V_{n}}G(x,z)-2\cdot 3^{-(n+1)}G(x,x)+\int _{B(x,2^{-(n+1)})}G(x,y)d\mu (y) \\&=\frac{2}{3}\cdot \frac{1}{5^{n}}\sum _{z\in V_{n}}g_{n}(x,z)+\int _{B(x,2^{-(n+1)})}\big (G(x,y)-G(x,x)\big )d\mu (y). \end{aligned}$$\end{document}Ibulk=2·3-(n+1)∑z∈Vn\V0,z≠xG(x,z)+∫B(x,2-(n+1))G(x,y)dμ(y)=2·3-(n+1)∑z∈VnG(x,z)-2·3-(n+1)G(x,x)+∫B(x,2-(n+1))G(x,y)dμ(y)=23·15n∑z∈Vngn(x,z)+∫B(x,2-(n+1))(G(x,y)-G(x,x))dμ(y).Above, we used the fact that G(x,z)=0G(x,z)=0 if one of x and z belongs to \documentclass[12pt]{minimal}
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\begin{document}$$V_{0}$$\end{document}V0, 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\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} 3\cdot 5^{n}&\int _{SG}G(x,y)d\mu (y)-2\sum _{y\in V_{n}}g_{n}(x,y)=3\cdot 5^{n}(I_{bulk}+I_{boundary})-2\sum _{y\in V_{n}}g_{n}(x,y)\\&=3\cdot 5^{n}\int _{B(x,2^{-(n+1)})}\big (G(x,y)-G(x,x)\big )d\mu (y)+3\cdot 5^{n}\sum _{z\in V_{0}}\int _{B(z,2^{-(n+1)})}G(x,y)d\mu (y). \end{aligned} \end{aligned}$$\end{document}3·5n∫SGG(x,y)dμ(y)-2∑y∈Vngn(x,y)=3·5n(Ibulk+Iboundary)-2∑y∈Vngn(x,y)=3·5n∫B(x,2-(n+1))(G(x,y)-G(x,x))dμ(y)+3·5n∑z∈V0∫B(z,2-(n+1))G(x,y)dμ(y).We claim that this expression is uniformly bounded in \documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}$$\end{document}n∈N and in \documentclass[12pt]{minimal}
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\begin{document}$$x\in V_{*}$$\end{document}x∈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 \documentclass[12pt]{minimal}
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\begin{document}$$B(x,2^{-(n+1})$$\end{document}B(x,2-(n+1) for all \documentclass[12pt]{minimal}
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\begin{document}$$x\in V_*\backslash V_0$$\end{document}x∈V∗\V0, \documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}$$\end{document}n∈N and \documentclass[12pt]{minimal}
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\begin{document}$$y\in B(x,2^{-(n+1)})$$\end{document}y∈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 \documentclass[12pt]{minimal}
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\begin{document}$$V_0$$\end{document}V0 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\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\max \bigg \{3\cdot 5^{n}\int _{B(x,2^{-(n+1)})}\big (G(x,x)-G(x,y)\big )d\mu (y),~3\cdot 5^{n}\sum _{z\in V_{0}}\int _{B(z,2^{-(n+1)})}G(x,y)d\mu (y)\bigg \} \\&\quad \le \max \bigg \{3\cdot 5^{n}\cdot 2\cdot 3^{-(n+1)}\cdot C\left( \frac{3}{5}\right) ^{n+1},~3\cdot 5^{n}\cdot 3\cdot 3^{-(n+1)}\cdot C\left( \frac{3}{5}\right) ^{n+1}\bigg \} \\&\quad =\max \bigg \{\frac{6}{5}C,~\frac{9}{5}C\bigg \}=\frac{9}{5}C, \end{aligned}$$\end{document}max{3·5n∫B(x,2-(n+1))(G(x,x)-G(x,y))dμ(y),3·5n∑z∈V0∫B(z,2-(n+1))G(x,y)dμ(y)}≤max{3·5n·2·3-(n+1)·C35n+1,3·5n·3·3-(n+1)·C35n+1}=max{65C,95C}=95C,thereby proving the above claim and hence the Lemma. □ □
Using our previous result, we can now directly prove (I1) in Theorem 1.1 by using the already established (I3).
Proposition 4.3
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\begin{document}$$g_n*\textsf{id}_n$$\end{document}gn∗idn grows asymptotically as \documentclass[12pt]{minimal}
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\begin{document}$$5^n$$\end{document}5n, and for all x∈ SGx∈SG it holds that\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{5^n}(g_n*\textsf{id}_n)^{\blacktriangle }(x)=4\int _{SG}G(x,y)d\mu (y). \end{aligned}$$\end{document}limn→∞15n(gn∗idn)▴(x)=4∫SGG(x,y)dμ(y).
Proof
Recall for every \documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}$$\end{document}n∈N the definition of the piecewise-constant continuatione of \documentclass[12pt]{minimal}
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\begin{document}$$u^\blacktriangle :SG\rightarrow \mathbb {R},u^\blacktriangle (x)=u(x_n)$$\end{document}u▴:SG→R,u▴(x)=u(xn), where \documentclass[12pt]{minimal}
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\begin{document}$$x_n\in V_n$$\end{document}xn∈Vn is the closest point to x∈ SGx∈SG in the n-th iteration. Let us introduce the shorthand notation \documentclass[12pt]{minimal}
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\begin{document}$$\beta _{n}:SG\rightarrow \mathbb {R}$$\end{document}βn:SG→R given by\documentclass[12pt]{minimal}
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\begin{document}$$ \beta _{n}(x){:}{=}\frac{1}{5^{n}}(g_{n}*id_{n})^{\blacktriangle }(x)-4\int _{SG}G(x,y)d\mu (y),$$\end{document}βn(x):=15n(gn∗idn)▴(x)-4∫SGG(x,y)dμ(y),which can be rewritten as\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \beta _{n}(x)&=\left( \frac{1}{5^{n}}(g_{n}*id_{n})^{\blacktriangle }(x)-4\int _{SG}(G(\cdot ,y))^{\blacktriangle }(x)d\mu (y)\right) \\&\quad +4\int _{SG}\big ((G(\cdot ,y))^{\blacktriangle }(x)-G(x,y)\big )d\mu (y) \\&{=}{:}D_{n}(x)+D_{n}^{\prime }(x). \end{aligned}$$\end{document}βn(x)=15n(gn∗idn)▴(x)-4∫SG(G(·,y))▴(x)dμ(y)+4∫SG((G(·,y))▴(x)-G(x,y))dμ(y)=:Dn(x)+Dn′(x).Using Proposition 4.2 and \documentclass[12pt]{minimal}
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\begin{document}$$\sup _{x\in SG}\textsf{d}(x)=\frac{1}{2}$$\end{document}supx∈SGd(x)=12 we get\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \limsup _{n\rightarrow \infty }|D_{n}(x)|&\le \limsup _{n\rightarrow \infty }\frac{1}{2^{n}}\left| (g_{n}*id_{n})^{\blacktriangle }(x)-4\cdot 5^{n}\int _{SG}(G(\cdot ,y))^{\blacktriangle }(x)d\mu (y)\right| \\&\quad \cdot \limsup _{n\rightarrow \infty }\frac{2^{n}}{5^{n}} = 0. \end{aligned}$$\end{document}lim supn→∞|Dn(x)|≤lim supn→∞12n(gn∗idn)▴(x)-4·5n∫SG(G(·,y))▴(x)dμ(y)·lim supn→∞2n5n=0.On the other hand, by Lemma 2.3, for every x∈ SGx∈SG we have\documentclass[12pt]{minimal}
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\begin{document}$$ \big |(G(\cdot ,y))^{\blacktriangle }(x)-G(x,y)\big |=|G(x_{n},y)-G(x,y)|\le R_{eff}(x_{n},x)\xrightarrow {n\rightarrow \infty }0,$$\end{document}|(G(·,y))▴(x)-G(x,y)|=|G(xn,y)-G(x,y)|≤Reff(xn,x)→n→∞0,which together with the dominated convergence theorem yields \documentclass[12pt]{minimal}
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\begin{document}$$\lim _{n\rightarrow \infty }D_{n}^{\prime }(x)=0$$\end{document}limn→∞Dn′(x)=0. Hence it holds \documentclass[12pt]{minimal}
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\begin{document}$$\lim _{n\rightarrow \infty }\beta _{n}(x)=0$$\end{document}limn→∞βn(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.
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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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All data supporting the findings are available in the supplementary material and in a public repository.
Demonstration content: the full text of this article is adapted from “On the structure of the sandpile identity element on Sierpiński gasket graphs” by Kaiser and colleagues (Computational & Applied Mathematics, 2026; https://doi.org/10.1007/s40314-026-03928-6), which is available under a Creative Commons license (https://creativecommons.org/licenses/by/4.0/).