Universality of boundary height probabilities in the two-dimensional Abelian sandpile beyond height zero: the Jeng–Piroux–Ruelle linear relations, lattice symmetry conditions, and mixed boundary conditions

The stationary two-dimensional Abelian sandpile model (ASM) has four height variables at each site (labelled 0–3 on the square lattice). Elvidge and Járai (2026) proved that on periodic lattices whose embedding is invariant under a rotation of order at least 3 about every vertex, the height-0 probability at a point $z$ of the lattice approximation (spacing $\varepsilon$) of a domain $U$ with open boundary conditions satisfies $p_{U,\varepsilon}(0;z)=p_{\mathcal{G}}(0)+\varepsilon^{2}c_{\mathcal{G}}f_{U}(z)+O(\varepsilon^{3})$, where $c_{\mathcal{G}}>0$ depends only on the lattice and $f_{U}$ is conformally covariant with scale dimension 2, equal to $2|\varphi_{U}'(z)|^{2}$ for simply connected $U$; they also showed by counterexamples that the symmetry assumption cannot be dropped. Their paper poses three open questions which together ask for the rigorous universality class of one-site height probabilities near boundaries and for a rigorous test of the Jeng–Piroux–Ruelle (2006) conjecture that the leading corrections of all four height probabilities satisfy linear relations in any geometry and with any boundary condition: (i) extend the scaling analysis to the height variables 1, 2, 3 — whose corrections are expected to carry logarithmic terms and whose bulk probabilities are not even rigorously known — in a form that enables this check; (ii) characterize precisely, or find a more general sufficient condition for, the local lattice symmetry required, and find the appropriate generalization to lattices (doubled-edge lattices, general critical isoradial graphs) where the conformal-radius formula fails; and (iii) extend the results to other boundary conditions, in particular boundaries that are partly open and partly closed, for which only conformal-field-theory predictions currently exist. No later work addresses these questions; the problem is open.

Background

The Abelian sandpile model (ASM) is a lattice model of self-organized criticality introduced by Bak, Tang and Wiesenfeld (1987) and given its abelian structure by Dhar (1990). On a finite connected graph with a distinguished sink vertex $s$, each non-sink vertex $x$ carries a number of particles; when $\eta(x)\ge\deg(x)$ the vertex topples, sending one particle along each incident edge, and particles reaching $s$ are lost. Adding particles at randomly chosen vertices and stabilizing defines an irreducible Markov chain whose stationary distribution $\nu_{V}$ is the object of interest. When $V$ is a finite subset of a two-dimensional lattice and the complement of $V$ is collapsed to the sink ("open boundary conditions"), the measures $\nu_{V}$ converge, as $V$ exhausts the lattice, to an infinite-volume stationary measure $\nu$. On the square lattice the height $\eta(x)$ takes values in $\{0,1,2,3\}$; many physics papers, including Jeng–Piroux–Ruelle (JPR), label the same four variables $1,2,3,4$, and this record follows the 0–3 labelling of Elvidge–Járai.

Majumdar and Dhar (1991, 1992) identified the structure of $\nu_{V}$ through the burning algorithm: recurrent configurations are in bijection with wired uniform spanning trees (UST), that is, spanning trees of the graph obtained by identifying the boundary with the sink. Under this burning bijection the event $\{\eta(o)=0\}$ is essentially the event that $o$ is a leaf of the wired tree, and the height-0 probability is a small determinant of the transfer-current matrix (the matrix of edge-pair Green-function differences whose determinants give edge-inclusion probabilities of the UST). This locality is what makes the height-0 variable tractable: Majumdar and Dhar computed the bulk value $p(0)=2(\pi-2)/\pi^{3}\approx0.0736$ on $\mathbb{Z}^{2}$ exactly. For heights $1,2,3$ the corresponding events are non-local — a neighbour of $o$ can fail to be a "predecessor" of $o$ either because the connecting tree edge is absent or because a long chain of tree edges separates the two sites (Ruelle 2021) — and even the bulk probabilities $p(i)$, $i=1,2$, on $\mathbb{Z}^{2}$ are known only through conjectured closed forms (JPR 2006), confirmed numerically to about $10^{-12}$ but unproven. General background on the rigorous theory can be found in the surveys of Járai (2018) and Ruelle (2021).

The scaling-limit physics of the model is governed by a logarithmic conformal field theory (a conformal field theory, CFT, whose fields can form Jordan cells, so that correlation functions contain logarithms) with central charge $c=-2$, representable in terms of symplectic fermions. Rigorously, Dürre (2009) proved conformal covariance of the height-one field on the whole plane — this is the field of the height variable labelled 1 in the 1–4 convention, i.e. height 0 here; Kassel and Wu (2015) proved conformal covariance of UST pattern fields; Chiarini, Cipriani, Rapoport and Ruszel (2023) and Rapoport (2023) developed the fermionic Gaussian free field structure of the ASM/UST, including joint distributions of the UST degree; and Adame-Carrillo and Ruszel (2025) identified the scaling limit of the fermionic discrete Gaussian free field with the symplectic-fermion CFT. All rigorous results to date concern the height-0 (equivalently, dissipative) sector; correlation functions involving heights $1,2,3$ remain conjectural.

Near a boundary, Brankov, Ivashkevich and Priezzhev (1993) computed on the square-lattice upper half plane $\mathbb{H}$ the leading correction to the height-0 probability at microscopic distance $m$ from an open boundary, of order $m^{-2}$. JPR (2006) computed, still on $\mathbb{H}$ but for all four height variables and for both open and closed boundaries, the full order-$m^{-2}$ asymptotics

$$p^{\mathrm{op}}_{m}(i)=p(i)+\frac{a_{i}+b_{i}\log m+b_{i}/2}{m^{2}}+o(m^{-2}),\qquad p^{\mathrm{cl}}_{m}(i)=p(i)-\frac{a_{i}+b_{i}\log m}{m^{2}}+o(m^{-2}),$$

with explicit constants $a_{i},b_{i}$ and $b_{0}=0$: only the height-0 variable has no logarithmic term. They observed that at this order the corrections for heights 2 and 3 are fixed linear combinations of those for heights 0 and 1, namely $p_{m}(2)-p(2)=\alpha_{3}[p_{m}(1)-p(1)]+\beta_{3}[p_{m}(0)-p(0)]$ and $p_{m}(3)-p(3)=\alpha_{4}[p_{m}(1)-p(1)]+\beta_{4}[p_{m}(0)-p(0)]$ with $\alpha_{3}=\frac{8-\pi}{2(\pi-2)}\approx2.12791$, $\beta_{3}=\frac{\pi^{3}-5\pi^{2}+12\pi-48}{4(\pi-2)^{2}}\approx-5.49453$, $\alpha_{4}=-\frac{\pi+4}{2(\pi-2)}\approx-3.12791$, $\beta_{4}=\frac{32+4\pi+\pi^{2}-\pi^{3}}{4(\pi-2)^{2}}\approx4.49453$ (these are JPR's heights 3 and 4 expressed through their heights 1 and 2). They conjectured that these linear relations hold "in any geometry and for any sort of boundary condition", and likewise for multisite probabilities, leaving heights 0 and 1 as the only independent one-site variables in the scaling limit. In the CFT picture the height-0 field is a primary field of scaling dimension 2 and the fields of heights 1, 2, 3 are its logarithmic partners; JPR also computed, at the CFT level, the one-point height functions in front of a boundary that is open except on a closed interval, using boundary-condition-changing operators (chiral boundary fields inserted at the points where the boundary condition switches from open to closed, here of conformal weight $-1/8$). Later physics work extended these computations: Poncelet and Ruelle (2017) treated multipoint correlators via a "zipper" spanning-forest formalism (the graph Laplacian is twisted by a complex connection along a semi-infinite defect line, allowing spanning forests with prescribed component connectivity to be counted), and Poncelet and Ruelle (2018) computed one-site probabilities on the triangular and hexagonal lattices, on the full plane and on half-planes, finding again the $(c+d\log r)/r^{2}$ form and verifying the open/closed and linear-relation structure wherever the computation could be carried out. Ruelle's 2021 review records that a lattice proof of the linear relations "remains one of the greatest challenges in the sandpile models".

Elvidge and Járai (2026) initiated the rigorous side of this program. Their Theorem 1.1 states: let $\mathcal{G}$ be a multigraph embedded in the plane in a $\mathbb{Z}^{2}$-periodic way, such that the embedding is invariant under a rotation by $2\pi/q_{v}$, $q_{v}\ge3$, around every vertex $v$ (which forces $q_{v}\in\{3,4,6\}$); let $U\subsetneq\mathbb{C}$ be a region whose boundary satisfies a mild accessibility condition, and let $U_{\varepsilon}$ be the lattice approximation with spacing $\varepsilon$ and open boundary conditions. Then

$$p_{U,\varepsilon}(0;z)=p_{\mathcal{G}}(0)+\varepsilon^{2}\,c_{\mathcal{G}}\,f_{U}(z)+O(\varepsilon^{3}),$$

where $p_{\mathcal{G}}(0)$ is the bulk height-0 probability, $c_{\mathcal{G}}>0$ depends only on $\mathcal{G}$ (e.g. $c_{\mathbb{Z}^{2}}=(\pi-2)/\pi^{3}$, which recovers the Brankov–Ivashkevich–Priezzhev half-plane result; $c_{\text{triangular}}=5\sqrt{3}/(24\pi)$), and $f_{U}$ is conformally covariant with scale dimension 2; for simply connected $U$, $f_{U}(z)=2|\varphi_{U}'(z)|^{2}$, that is, twice the squared inverse conformal radius $\mathrm{rad}(U,z)=|\varphi_{U}'(z)|^{-1}$, where $\varphi_{U}:U\to\mathbb{D}$ is any conformal map with $\varphi_{U}(z)=0$. Adame-Carrillo and Ruszel (2025, Example 4.10) independently proved this formula for the square lattice and simply connected regions via the fermionic discrete Gaussian free field. Crucially, Elvidge and Járai show that the symmetry assumption cannot be dropped: (a) on the lattice with each horizontal edge doubled (only 180-degree symmetric), even after the vertical stretch ($\sqrt2$) that makes the random walk isotropic, the $\varepsilon^{2}$ coefficient is not conformally covariant (the relevant coefficient ratio is $\approx1.3475$), but covariance is restored by a different, larger stretch $b\approx1.6417$ that makes the walk anisotropic — a phenomenon for which the authors state they have no intuitive understanding; (b) on general critically weighted isoradial graphs (graphs embedded so that every face is inscribed in a circle of common radius, with edge conductances equal to the tangent of the rhombus half-angle, where Kenyon (2002) and Chelkak and Smirnov (2011) proved isotropy of the scaling limit of the walk), the natural analogue of the theorem fails. This is the "partial universality" of their title, and it is the state of the art against which the present problem is posed.

Problem Statement

Complete the rigorous universality picture of one-site height probabilities of the two-dimensional Abelian sandpile near boundaries, of which Elvidge and Járai's Theorem 1.1 is the height-0 first step, by answering the three questions their paper poses. The common objective uniting them is to make the Jeng–Piroux–Ruelle linear-relations conjecture rigorous at the one-site level — or to delimit it rigorously. Throughout, heights are labelled 0–3, $U_{\varepsilon}$ is the lattice approximation of a region $U$ with spacing $\varepsilon$ and open boundary conditions unless stated otherwise, and $\mathcal{G}$ is a periodic lattice satisfying the hypotheses of Theorem 1.1 or whatever weaker hypothesis is established under question 2.

1. Heights 1–3. Establish the order-$\varepsilon^{2}$ asymptotics of $p_{U,\varepsilon}(i;z)$ for $i=1,2,3$ in general domains, including the logarithmic terms expected from the non-vanishing coefficients $b_{i}$ (on the upper half plane JPR found $p^{\mathrm{op}}_{m}(i)=p(i)+m^{-2}(a_{i}+b_{i}\log m+b_{i}/2)+o(m^{-2})$), in a form that permits a rigorous check of the JPR conjecture: at order $\varepsilon^{2}$, the corrections $p_{U,\varepsilon}(2;z)-p_{\mathcal{G}}(2)$ and $p_{U,\varepsilon}(3;z)-p_{\mathcal{G}}(3)$ must be the fixed linear combinations, with coefficients $\alpha_{3},\beta_{3},\alpha_{4},\beta_{4}$ as computed by JPR, of $p_{U,\varepsilon}(1;z)-p_{\mathcal{G}}(1)$ and $p_{U,\varepsilon}(0;z)-p_{\mathcal{G}}(0)$. Proving or refuting these relations in the generality in which the height-0 result holds is the target. Since even the bulk constants $p_{\mathcal{G}}(i)$, $i=1,2$, are currently only conjectured, a solution may need to establish these constants or to formulate the comparison through suitably renormalized lattice expressions.

2. Symmetry condition. Give a precise characterization — or at least a strictly more general sufficient condition than invariance under a rotation of order at least 3 about every vertex — of the local symmetry of the periodic lattice under which the conclusion of Theorem 1.1 holds; and find the appropriate generalization of the theorem to lattices where it fails. In particular: decide whether the rotational-symmetry hypothesis is essentially optimal; identify what replaces the conformal-radius function $f_{U}(z)=2|\varphi_{U}'(z)|^{2}$ on the lattice with doubled horizontal edges and on general critically weighted isoradial graphs; and account for the anomalous scaling (the vertical stretch $b\approx1.6417$, larger than the isotropic value $\sqrt2$) that restores conformal covariance on the doubled-edge lattice for both the one-point and the two-point height-0 functions. A strictly more general sufficient condition settles the characterization of the symmetry class; deciding whether the rotational-symmetry hypothesis is essentially optimal and identifying what replaces $f_{U}$ on the doubled-edge and isoradial lattices are separate required sub-results.

3. Boundary conditions. Extend the scaling results — for height 0, and, per question 1, for heights 1–3 — to other boundary conditions, for example when part of the boundary is open and part is closed. This includes constructing the relevant infinite-volume stationary measures with partially closed boundaries where needed, and comparing the result with the conformal-field-theory prediction of JPR for mixed boundaries (via boundary-condition-changing operators of weight $-1/8$), which currently has no rigorous lattice counterpart. The mixed open/closed boundary is the required target; more general boundary conditions are a bonus.

A complete answer determines the universality class of one-site height probabilities near boundaries: which height variables, which lattices and local symmetries, and which boundary conditions exhibit conformally covariant $\varepsilon^{2}$ scaling satisfying the JPR linear relations, and what replaces this structure outside that class. The three components are jointly held together by this objective; resolving only one of them is partial progress.

References

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