Scaling and partial universality of the height zero probability in the 2D Abelian sandpile

Abstract

We study the extent of universality in the scaling of the height $0$ probability of the stationary Abelian sandpile on lattice approximations of a region $U \subset \mathbb{C}$ with lattice spacing $\varepsilon$ and with open boundary conditions. We show that under certain symmetry assumptions on the lattice, in the scaling limit $\varepsilon \to 0$ this probability at a point $z \in U$ equals $p_{\mathcal{G}}(0) + \varepsilon^2 c_{\mathcal{G}} f_U(z) + O(\varepsilon^3)$, where $p_{\mathcal{G}}(0)$ is the height $0$ probability on the full lattice, $c_{\mathcal{G}} > 0$ is a constant depending only on the lattice, and $f_U$ is a conformally covariant positive real-valued function depending only on $U$. This generalises a result of Brankov, Ivashkevich and Priezzhev (1993), who considered the square lattice and the upper half plane. It also generalizes Example 4.10 in the recent study of the fermionic DGFF of Adame-Carillo and Ruszel (2025). We show via counterexamples that our symmetry assumption cannot be omitted, although conformal covariance may be recovered via an unusual scaling. In particular, the natural analogue of our result fails on general isoradial graphs. Our work is motivated by Jeng, Piroux and Ruelle (2006), who computed the correction terms for all height variables in the special case of the upper half plane in $\mathbb{Z}^2$. They conjectured that their results hold more generally and the present paper is a step in an attempt to make this conjecture rigorous.