On the algebraic area of cubic lattice walks

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.

arXiv comment14 pages, 5 figures, comments welcome

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Cited by in corpus (2)

• Walking on Archimedean Lattices: Insights from Bloch Band Theory
• Lattice random walks and quantum A-period conjecture