On the algebraic area of cubic lattice walks

TL;DR

This paper derives an explicit enumeration formula for closed cubic lattice walks with a given length and 3D algebraic area, linking it to quantum exclusion statistics and cluster coefficients.

Contribution

It introduces a novel explicit formula for counting cubic lattice walks with specified 3D algebraic area, connecting combinatorics with quantum statistical mechanics.

Findings

Explicit enumeration formula for cubic lattice walks

Mapping to quantum exclusion statistics cluster coefficients

Constraint linking fermionic particle types

Abstract

We obtain an explicit formula to enumerate closed random walks on a cubic lattice with a specified length and 3D algebraic area. The 3D algebraic area is defined as the sum of algebraic areas obtained from the walk's projection onto the three Cartesian planes. This enumeration formula can be mapped onto the cluster coefficients of three types of particles that obey quantum exclusion statistics with statistical parameters $g=1$, $g=1$, and $g=2$, respectively, subject to the constraint that the numbers of $g=1$ (fermions) exclusion particles of two types are equal.