Lattice random walks and quantum A-period conjecture

TL;DR

This paper derives explicit formulas for counting classical lattice walks with signed area and connects this enumeration to quantum periods in topological string theory, revealing deep links between combinatorics, physics, and geometry.

Contribution

It introduces a novel connection between lattice walk enumeration and quantum A-period conjecture in topological string theory, providing explicit formulas and a unifying framework.

Findings

Explicit formulas for $C_N(A)$ on square and triangular lattices.

Mapping of enumeration problem to Hofstadter-like Hamiltonian trace.

Conjectured link between signed area enumeration and quantum A-period in string theory.

Abstract

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 $\mathcal{B}_3$.