AI 中文总结
该论文研究二维阿贝尔沙堆高度为0的概率的标度普适性，在满足对称性的格点上推导其标度极限表达式，推广了前人结果，通过反例说明对称性假设的必要性，为严格化Jeng等人的推测提供了基础。
AI 中文摘要
我们研究了在复平面区域U的格点近似上，具有格点间距ε和开放边界条件的稳态阿贝尔沙堆中高度为0的概率的标度性的普适程度。我们证明，在格点满足一定对称性假设的情况下，当标度极限ε→0时，区域U内某点z处的该概率等于p_G(0) + ε² c_G f_U(z) + O(ε³)，其中p_G(0)是完整格点上高度为0的概率，c_G>0是仅依赖于格点的常数，f_U是仅依赖于U的保角协变正实值函数。这一结果推广了Brankov、Ivashkevich和Priezzhev（1993）的研究，他们当时考虑的是正方形格点和上半平面的情况；同时也推广了Adame-Carillo和Ruszel（2025）关于费米子DGFF的近期研究中的例4.10。我们通过反例证明，我们的对称性假设不能省略，尽管保角协变性可以通过一种特殊的标度恢复；特别是，我们的结果的自然类似物在一般等距格图上不成立。本研究的动机来自Jeng、Piroux和Ruelle（2006）的工作，他们计算了Z²中特殊上半平面情况下所有高度变量的修正项，他们推测其结果可更广泛适用，本文正是为将该推测严格化所做的一步工作。
英文摘要
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.
Comments36 pages, 4 figures
