Meanders: exact asymptotics

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 = root 29 (root 29 + root 5) / 12, and the semi-meander exponent <(alpha)over bar> = 1 root 11(root 29 + root 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. (C) 2000 Elsevier Science B.V. All rights reserved.