Meanders: Exact Asymptotics

TL;DR

This paper proposes a theoretical framework linking meanders to conformal field theory, providing exact predictions for their exponents and generalizing results to various geometries and loop parameters.

Contribution

It introduces a novel conformal field theory approach to exactly predict meander exponents and extends the analysis to different geometries and loop fugacities.

Findings

Exact formulas for meander exponents nd ar{\u03b1}

Agreement with recent numerical estimates

Generalization to various geometries and loop fugacities

Abstract

We conjecture that meanders are governed by the gravitational version of a c=-4 two-dimensional conformal field theory, allowing for exact predictions for the meander configuration exponent \alpha=\sqrt{29}(\sqrt{29}+\sqrt{5})/12, and the semi-meander exponent {\bar\alpha}=1+\sqrt{11}(\sqrt{29}+\sqrt{5})/24. This result follows from an interpretation of meanders as pairs of fully packed loops on a random surface, described by two c=-2 free fields. The above values agree with recent numerical estimates. We generalize these results to a score of meandric numbers with various geometries and arbitrary loop fugacities.