2024/12/30 by Li Gan, Gan, Li · 1 citation
Computer Science · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2412.21128
openalex publication_date 2024/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive explicit closed-form expressions for the generating function CN(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 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 \mathbbF0 geometry, while triangular lattice walks are associated with local B3.