Prudent walks and polygons

TL;DR

This paper extends the enumeration of prudent walks and polygons to two and three dimensions, analyzing their growth constants and critical exponents using transfer matrix algorithms.

Contribution

It introduces extended series for prudent polygons in 2D and 3D, and investigates their growth constants and critical exponents across different classes.

Findings

Growth constant for polygons is smaller than for walks in 2D.

Polygon growth constants vary with class, walk constants do not.

Estimated critical exponents in 3D for walks and polygons.

Abstract

We have produced extended series for two-dimensional prudent polygons, based on a transfer matrix algorithm of complexity O$(n^5),$ for a series of length $n.$ We have extended the definition to three dimensions and produced series expansions for both prudent walks and polygons in three dimensions. For prudent polygons in two dimensions we find the growth constant to be smaller than that for the corresponding walks, and by considering three distinct classes of polygons, we find that the growth constant for polygons varies with class, while for walks it does not. We give the critical exponent for both walks and polygons. In the three-dimensional case we estimate the growth constant for both walks and polygons and also estimate the usual critical exponents $\gamma,$ $\nu$ and $\alpha.$