Meanders: A Direct Enumeration Approach

TL;DR

This paper investigates semi-meander configurations as a model for polymer folding, using enumeration and q-expansion techniques to analyze winding behavior and predict a phase transition at q=2.

Contribution

It introduces a direct enumeration method for semi-meanders and reformulates data into a q-expansion to study winding transitions.

Findings

Predicted a transition at q=2 between winding regimes.

Performed enumeration up to n=29 for statistical analysis.

Reformulated data into a large q expansion for theoretical insights.

Abstract

We study the statistics of semi-meanders, i.e. configurations of a set of roads crossing a river through n bridges, and possibly winding around its source, as a toy model for compact folding of polymers. By analyzing the results of a direct enumeration up to n=29, we perform on the one hand a large n extrapolation and on the other hand we reformulate the available data into a large q expansion, where q is a weight attached to each road. We predict a transition at q=2 between a low-q regime with irrelevant winding, and a large-q regime with relevant winding.