Families of prudent self-avoiding walks

TL;DR

This paper analyzes various classes of prudent self-avoiding walks on lattices, providing exact enumeration for some classes, revealing complex generating functions, and studying their asymptotic behavior.

Contribution

It exactly solves the enumeration of the third class of prudent walks and introduces an isotropic family on the triangular lattice, expanding understanding of these complex structures.

Findings

Third class generating function is not algebraic or D-finite.

Exact enumeration of an isotropic family on the triangular lattice.

Endpoint of walks moves away from origin at positive speed.

Abstract

A self-avoiding walk (SAW) on the square lattice is prudent if it never takes a step towards a vertex it has already visited. Prudent walks differ from most classes of SAW that have been counted so far in that they can wind around their starting point. Their enumeration was first addressed by Pr\'ea in 1997. He defined 4 classes of prudent walks, of increasing generality, and wrote a system of recurrence relations for each of them . However, these relations involve more and more parameters as the generality of the class increases. The first class actually consists of partially directed walks, and its generating function is well-known to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (2005). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even D-finite. The fourth class -- general prudent walks -- is the only isotropic one, and still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-D-finite. We also study the asymptotic properties of these classes of walks, with the (somewhat disappointing) conclusion that their endpoint moves away from the origin at a positive speed. This is confirmed visually by the random generation procedures we have designed.