Geometrically constrained statistical systems on regular and random lattices: From folding to meanders

TL;DR

This paper reviews recent advances in two-dimensional statistical models with geometric constraints, focusing on folding problems, meanders, and their reformulations as fully packed loop models with Coulomb gas descriptions.

Contribution

It provides a unified Coulomb gas framework for various constrained models and presents exact results and conjectures, including the meander configuration exponent.

Findings

Exact results for folding and meander problems

Unified Coulomb gas description of constrained models

Derivation of the meander configuration exponent

Abstract

We review a number a recent advances in the study of two-dimensional statistical models with strong geometrical constraints. These include folding problems of regular and random lattices as well as the famous meander problem of enumerating the topologically inequivalent configurations of a meandering road crossing a straight river through a given number of bridges. All these problems turn out to have reformulations in terms of fully packed loop models allowing for a unified Coulomb gas description of their statistical properties. A number of exact results and physically motivated conjectures are presented in detail, including the remarkable meander configuration exponent alpha=(29+sqrt(145))/12.