Lattice random walks and quantum A-period conjecture

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.