Algebraic area enumeration for open lattice walks

TL;DR

This paper extends an operator method to enumerate open lattice walks with fixed length and algebraic area, including walks with specific endpoint constraints, providing a new approach for open walk area counting.

Contribution

It introduces an extension of the operator method for counting open lattice walks with algebraic area, including fixed endpoints and specific boundary conditions.

Findings

Derived formulas for open walk enumeration with algebraic area

Extended operator method to open walks with fixed endpoints

Outlined approach for walks with fully fixed endpoints

Abstract

We calculate the number of open walks of fixed length and algebraic area on a square planar lattice by an extension of the operator method used for the enumeration of closed walks. The open walk area is defined by closing the walks with a straight line across their endpoints and can assume half-integer values in lattice cell units. We also derive the length and area counting of walks with endpoints on specific straight lines and outline an approach for dealing with walks with fully fixed endpoints.