标题： 格子随机游走与量子A周期猜想 显示英文标题

标题： Lattice random walks and quantum A-period conjecture

摘要： 我们推导出生成函数$C_N(A)$的显式闭合表达式，该生成函数对在方形和三角形格点上的经典封闭随机行走进行计数，这些行走具有$N$步且带有符号面积$A$，其特征由每个跳跃方向的移动次数决定。 这一计数问题被映射到各向异性霍夫施塔特（Hofstadter）类哈密顿量幂次的迹，并与排除粒子的聚类系数相关：方形格点行走的排除强度参数$g = 2$，以及三角形格点行走的$g = 1$和$g = 2$的混合。 通过利用霍夫施塔特模型与高能物理之间的内在联系，我们提出了一种猜想，将统计力学中的上述符号面积计数$C_N(A)$与拓扑弦理论中相关 toric Calabi-Yau 三维流形的量子 A-周期联系起来：方形格子行走对应局部$\mathbb{F}_0$几何，而三角形格子行走则与局部$\mathcal{B}_3$相关。 显示英文摘要

摘要： We derive explicit closed-form expressions for the generating function $C_N(A)$, which enumerates classical closed random walks on square and triangular lattices with $N$ steps and a signed area $A$, characterized by the number of moves in each hopping direction. This enumeration problem is mapped to the trace of powers of anisotropic Hofstadter-like Hamiltonian and is connected to the cluster coefficients of exclusion particles: exclusion strength parameter $g = 2$ for square lattice walks, and a mixture of $g = 1$ and $g = 2$ for triangular lattice walks. By leveraging the intrinsic link between the Hofstadter model and high energy physics, we propose a conjecture connecting the above signed area enumeration $C_N(A)$ in statistical mechanics to the quantum A-period of associated toric Calabi-Yau threefold in topological string theory: square lattice walks correspond to local $\mathbb{F}_0$ geometry, while triangular lattice walks are associated with local $\mathcal{B}_3$.