Lattice random walks and quantum A-period conjecture

We derive explicit closed-form expressions for the generating function CN (A) CN (A), which enumerates classical closed random walks on square and triangular lattices with N N steps and a signed area A 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 g=2 for square lattice walks, and a mixture of g = 1 g=1 and g = 2 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 CN (A) CN (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 F₀ 𝔽0 geometry, while triangular lattice walks are associated with local B₃ ℬ3.