60度平行四边形上的Cardy公式类比：一种模方法
Analogue of Cardy's formula on the 60-degree parallelogram: a modular approach
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中文总结 AI 辅助
该研究将矩形上的Cardy公式类比到60度平行四边形，推导了Cardy-Smirnov函数的共形映射与模形式积分公式，并证明其唯一性，完善了临界渗流相关的模理论结果。
中文摘要 AI 辅助
四标记平面区域的Cardy-Smirnov函数编码了临界渗流穿越概率的猜想标度极限。Cardy预测了矩形上位点渗流和键渗流两种情况的该极限的闭式公式，Smirnov证明了三角格上临界位点渗流的该公式及其共形不变性。Kleban和Zagier随后表明，矩形上的该函数具有模解释：它由模函数方程结合温和的解析假设确定。我们在π/3平行四边形上开展了类比研究，推导了Cardy-Smirnov函数作为不完全贝塔积分的闭式共形映射公式，证明了将其实现为模形式η(τ)²η(3τ)²积分的闭式模公式，建立了唯一性定理，表明单个函数方程结合q展开假设和非退化条件可唯一确定该函数。
英文摘要
The Cardy-Smirnov function of a four-marked planar domain encodes the conjectural scaling limit of the crossing probability for critical percolation. Cardy predicted a closed formula for this limit on the rectangle in the cases of both site and bond percolation, and Smirnov proved the formula together with conformal invariance for critical site percolation on the triangular lattice. Kleban and Zagier later showed that on the rectangle the function admits a modular interpretation: it is determined by a modular functional equation together with a mild analytic ansatz. We carry out the analogue on the $π/3$ parallelogram. We derive a closed conformal-map formula for the Cardy-Smirnov function as an incomplete beta integral, prove a closed modular formula realizing it as an integral of the modular form $η(τ)^2η(3τ)^2$, and establish a uniqueness theorem showing that a single functional equation, together with a $q$-expansion ansatz and a nondegeneracy condition, determines the function uniquely.
