TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.19.2.053
TI  - Lattice random walks and quantum A-period conjecture
PY  - 2025/08/21
UR  - https://www.scipost.org/SciPostPhys.19.2.053
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 19
IS  - 2
SP  - 053
A1  - Gan, Li
AB  - 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 $\mathcal{B}_3$.
ER  -

@Article{10.21468/SciPostPhys.19.2.053,
	title={{Lattice random walks and quantum A-period conjecture}},
	author={Li Gan},
	journal={SciPost Phys.},
	volume={19},
	pages={053},
	year={2025},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.19.2.053},
	url={https://scipost.org/10.21468/SciPostPhys.19.2.053},
}