Contributor info: Dr Li Gan

Li Gan

SciPost Phys. 19, 053 (2025) · published 21 August 2025 | Toggle abstract · pdf

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 $\mathcal{B}_3$.