Meanders: A direct enumeration approach

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.