Weakly directed self-avoiding walks

TL;DR

This paper introduces weakly directed self-avoiding walks on the square lattice, analyzes their generating function, growth constant, and end-to-end distance, revealing complex singularity structure and growth behavior.

Contribution

It defines a new class of self-avoiding walks, characterizes their generating function, and studies their combinatorial and geometric properties.

Findings

Generating function has complex singularity structure and is not D-finite.

Growth constant is approximately 2.54, larger than many known SAW families.

End-to-end distance grows linearly.

Abstract

We define a new family of self-avoiding walks (SAW) on the square lattice, called weakly directed walks. These walks have a simple characterization in terms of the irreducible bridges that compose them. We determine their generating function. This series has a complex singularity structure and in particular, is not D-finite. The growth constant is approximately 2.54 and is thus larger than that of all natural families of SAW enumerated so far (but smaller than that of general SAW, which is about 2.64). We also prove that the end-to-end distance of weakly directed walks grows linearly. Finally, we study a diagonal variant of this model.