环面上的同调渗流：plaquettes 与 permutohedra
Homological percolation on a torus: plaquettes and permutohedra
AI总结：
本文构建环面上的高维同调渗流模型，证明巨型同调循环存在尖锐相变，并在中间维数得到临界阈值 $p_c=1/2$，推广 Kesten 的经典结果。
摘要：
我们研究正方形格点上键渗流和三角形格点上座渗流的高维同调类比。通过将某些无限胞腔复形按不断增长的子格取商，我们得到具有高度对称性且具有环面 $\mathbb{T}^d$ 拓扑结构的有限胞腔复形。当随机子复形在环境环面的同调中诱导出非平凡的 $i$ 维循环时，我们称此类循环为“巨型”循环。我们证明，对于每个 $i$ 和 $d$，都存在从巨型循环不存在到巨型循环张成环面同调的尖锐转变。我们还证明在某些情形下阈值函数收敛到一个常数。特别地，我们证明对于两种模型，在中间维数 $i=d/2$ 的情形下 $p_c=1/2$。这给出了 Kesten 关于正方形格点上键渗流和三角形格点上座渗流 $p_c=1/2$ 定理的有限体积高维类比。
英文摘要：
We study higher-dimensional homological analogues of bond percolation on a square lattice and site percolation on a triangular lattice. By taking a quotient of certain infinite cell complexes by growing sublattices, we obtain finite cell complexes with a high degree of symmetry and with the topology of the torus $\mathbb{T}^d$. When random subcomplexes induce nontrivial $i$-dimensional cycles in the homology of the ambient torus, we call such cycles \emph{giant}. We show that for every $i$ and $d$ there is a sharp transition from nonexistence of giant cycles to giant cycles spanning the homology of the torus. We also prove convergence of the threshold function to a constant in certain cases. In particular, we prove that $p_c=1/2$ in the case of middle dimension $i=d/2$ for both models. This gives finite-volume high-dimensional analogues of Kesten's theorems that $p_c=1/2$ for bond percolation on a square lattice and site percolation on a triangular lattice.
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