论文标题
圆环上的同源渗透：plaquettes和persutohedra
Homological percolation on a torus: plaquettes and permutohedra
论文作者
论文摘要
我们研究了在三角形晶格上的平方晶格和位点渗透上键渗透的较高维度同源类似物。 通过通过生长的sublattices来对某些无限细胞复合物进行某些无限细胞复合物，我们获得了具有高度对称性的有限细胞复合物，并且具有圆环$ \ mathbb {t}^d $的拓扑结构。当随机子复合物在环境圆环的同源性中诱导非平凡$ i $维循环时，我们称之为这种循环\ emph {giant}。我们表明，对于每$ i $和$ d $，从巨型周期不存在的急剧过渡到跨越圆环同源的巨型周期。 在某些情况下，我们还证明了阈值函数与常数的收敛性。特别是，我们证明，在中间维度$ i = d/2 $的情况下，$ p_c = 1/2 $。这给出了Kesten定理的有限体积的高维类似物，即$ P_C = 1/2 $用于在三角形晶格上的方形晶格和现场渗透上的键渗透。
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.
