AI 中文总结
本文研究双曲三角剖分上键渗流的环含量，发现总持久同调随曲率近线性变化，并推导出无自由参数的精确公式，与数值测量吻合。
AI 中文摘要
我们研究规则 $\{3,q\}$ 三角剖分（$q\ge7$）的圆盘形补丁上渗流簇的同调。改变 $q$ 给出一个单参数双曲剖分族，从轻度弯曲（$q=7$）到深度双曲（$q=20$），本文测量并推导了作为曲率函数的环含量。一个单一的生长率参数 $\lambda(q)$，即连续环尺寸的渐近比，支配所有结果。我们测量了 $q=7,\dots,20$ 时键渗流复形每个顶点的总持久 $H_1$，发现它是 $1/\lambda(q)$ 的近线性函数。一个精确恒等式将总环持久性归结为格点最小生成树的权重减去一个边界修正。边界修正由环递归得出，而生成树权重通过将环逐一附加到收缩的内部，并对通过外部环闭合的环进行修正，推导到 $1/\lambda(q)$ 的一阶。所得公式无自由参数，以大曲率截距的闭式形式给出 $5/4-2\pi/(3\sqrt3)$，并与测量结果在下一阶项的大小范围内一致。
英文摘要
We study the homology of percolation clusters on disk-shaped patches of the regular $\{3,q\}$ triangle tilings for $q\ge7$. Varying $q$ gives a one-parameter family of hyperbolic tilings, from mildly curved ($q=7$) to deeply hyperbolic ($q=20$), and this paper measures and derives loop content as a function of curvature. A single growth-rate parameter $λ(q)$, the asymptotic ratio of successive ring sizes, governs every result. Total persistent $H_1$ of the bond-percolation complex, per vertex, is measured across $q=7,\dots,20$ and found to be a nearly linear function of $1/λ(q)$. An exact identity reduces total loop persistence to the weight of the lattice's minimum spanning tree minus a boundary correction. The boundary correction follows from the ring recursion, and the spanning-tree weight is derived to first order in $1/λ(q)$ by attaching rings one at a time to a contracted interior, with a correction for loops that close through the ring outside. The resulting formula has no free parameters, gives the large-curvature intercept in closed form as $5/4-2π/(3\sqrt3)$, and agrees with the measurements to within the size of the next-order term.
Comments12 pages, 3 figures, code at https://github.com/ztreisman/curvature-percolation-homology
