AI 中文总结
研究不同随机化程序生成的四边形曲面统计性质，在统一框架构建曲面并建立操作关系。分析GKNS、DQ和GQ，确定相关性质及标度，通过多种统计量揭示它们不同特征，为非临界弦理论提供几何框架。
AI 中文摘要
我们研究了由不同随机化程序生成的随机四边形曲面的统计性质，包括Gruzberg-Klümper-Nuding-Sedrakyan（GKNS）构造以及动态三角剖分（DT）的两种新推广：动态四边形剖分（DQ）和通用四边形剖分（GQ）。我们在统一的图论框架中构建这些曲面，并建立定义不同系综的基本操作之间的关系。对于GKNS曲面，证明其构造等同于两个相互约束的渗流过程并确定相关临界点和临界指数，揭示与普通渗流的偏差。对于DQ和GQ，分析基础马尔可夫链并确定混合和弛豫时间的标度。通过度分布、度相关性、距离统计和豪斯多夫维数进一步分析这三个系综。DQ呈现指数衰减的度分布和与DT相似的几何性质，而GKNS和GQ显示宽泛的无标度度分布。此外，GKNS曲面具有渐近豪斯多夫维数$\Delta_H\approx 2$，而DQ和GQ接近$\Delta_H\approx 4$，与DT类似。这表明DQ与DT的普遍行为兼容，而GKNS和GQ定义了不同类别的随机几何，暗示随机曲面空间中不同的基础测度以及一类新的非临界弦理论可能的几何框架。
英文摘要
We investigate the statistical properties of random quadrangular surfaces generated by different randomization procedures: the Gruzberg-Klümper-Nuding-Sedrakyan (GKNS) construction, and two newly introduced generalizations of dynamical triangulations (DT), dynamical (DQ) and general quadrangulations (GQ). We formulate these surfaces within a unified graph-theoretic framework and establish the relationships between the elementary operations defining the different ensembles. For GKNS surfaces, we demonstrate that the construction is equivalent to two mutually constrained percolation processes and determine the associated critical point and critical exponents, revealing deviations from ordinary percolation. For DQ and GQ, we analyze the underlying Markov chains and determine the scaling of mixing and relaxation times. We further analyze all three ensembles through their degree distributions, degree correlations, distance statistics, and Hausdorff dimensions. While DQ exhibits exponentially decaying degree distributions and geometric properties similar to DT, GKNS and GQ display broad, scale-free degree distributions. Moreover, GKNS surfaces possess an asymptotic Hausdorff dimension $Δ_H\approx 2$, whereas DQ and GQ approach $Δ_H\approx 4$, similarly to DT. This indicates that DQ is compatible with the universal behavior of DT, while GKNS and GQ define distinct classes of random geometry, implying a different underlying measure in the space of random surfaces and a possible geometric framework for a new class of noncritical string theories.
