AI 中文总结
本文针对三维双曲空间中的单模随机无限 trivalent 双曲多面体，建立共形类型的统一几何概率理论，通过局部几何量的期望给出其全局共形类型的二分判据，还证明了有限多面体的 Benjamini-Schramm 极限的抛物性并得到面随机游走的双曲速度结论。
AI 中文摘要
我们针对三维双曲空间$\boldsymbol{\rm H}^3$中的单模随机无限 trivalent 双曲多面体，建立了共形类型的统一几何与概率理论。通过将这类多面体与对偶带角圆盘三角剖分及正则圆模式对应起来，我们为每个面对赋予了一个完全由局部二面角几何决定的内在几何特征数$L_f(P)$。对于根面$f$，我们建立了单模高斯-博内公式：$\boldsymbol{\rm E}[L_f(P)] = 2\pi - (\pi/3)\boldsymbol{\rm E}[\deg(f)]$。在自然的驯服性与可容许性假设下，这一公式导出了一个严格的二分结论：当$\boldsymbol{\rm E}[L_f(P)] = 0$时，单模随机 trivalent 双曲多面体是抛物型的；当$\boldsymbol{\rm E}[L_f(P)] < 0$时，则为双曲型的。由此可见，全局共形类型由局部几何量的期望决定。我们还研究了用有限多面体逼近无限多面体的问题，证明所有均匀面根有限 trivalent 双曲多面体的可容许 Benjamini-Schramm 极限必然是抛物型的，这揭示了双曲型单模多面体极限存在的几何与拓扑障碍。为研究双曲 regime 下的随机行为，我们克服了经典圆填充工具在无界度数下的失效问题，建立了正则圆模式的精细环引理，该引理通过局部花度数实现了对相邻圆半径的有效指数控制。结合边界方法，我们将泊松边界与无穷远圆等同，并证明了面随机游走具有正双曲速度。这些结果首次提供了将三维局部二面角几何、全局共形类型及单模随机无限双曲多面体的渐近随机行为关联起来的定量框架。
英文摘要
We develop a unified geometric and probabilistic theory of conformal type for unimodular random infinite trivalent hyperbolic polyhedra in $\mathbb{H}^3$. By corresponding these with dual angled disk triangulations and regular circle patterns, we associate to each face an intrinsic geometric characteristic number $L_f(P)$, determined entirely by local dihedral geometry. For the root face $f$, we establish the unimodular Gauss-Bonnet formula $\mathbb{E}[L_f(P)] = 2π- (π/3)\mathbb{E}[deg(f)]$. Under natural tameness and admissibility assumptions, this yields a sharp dichotomy: a unimodular random trivalent hyperbolic polyhedron is parabolic precisely when $\mathbb{E}[L_f(P)] = 0$, and hyperbolic when $\mathbb{E}[L_f(P)] < 0$. Thus, global conformal type is governed by the expectation of a local geometric quantity. We also investigate the approximation of infinite polyhedra by finite ones. We prove that every admissible Benjamini-Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra is necessarily parabolic, revealing a geometric and topological obstruction to the existence of hyperbolic unimodular polyhedral limits. To study stochastic behavior in the hyperbolic regime, we overcome the failure of classical circle packing tools for unbounded degrees by establishing a refined ring lemma for regular circle patterns. This yields effective exponential control of adjacent circle radii via local flower degrees. Combined with boundary methods, we identify the Poisson boundary with the circle at infinity and prove positive hyperbolic speed for the face random walk. These results provide the first quantitative framework connecting local three-dimensional dihedral geometry, global conformal type, and asymptotic stochastic behavior of unimodular random infinite hyperbolic polyhedra.
