The algebraic area of closed lattice random walks

TL;DR

This paper introduces a formula for counting closed lattice random walks of a given length that enclose a specific algebraic area, linking combinatorial enumeration with quantum physics models.

Contribution

It provides a novel explicit combinatorial formula for algebraic area enumeration of lattice walks, connecting it to Hofstadter's model via Kreft coefficients.

Findings

Derived a formula for algebraic area enumeration of lattice walks.

Connected combinatorial enumeration to Hofstadter's quantum model.

Split the enumeration problem into manageable combinatorial pieces.

Abstract

We propose a formula for the enumeration of closed lattice random walks of length $n$ enclosing a given algebraic area. The information is contained in the Kreft coefficients which encode, in the commensurate case, the Hofstadter secular equation for a quantum particle hopping on a lattice coupled to a perpendicular magnetic field. The algebraic area enumeration is possible because it is split in $2^{n/2-1}$ pieces, each tractable in terms of explicit combinatorial expressions.