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 α=√29(√29+√5)/12, and the semi-meander exponent α-bar=1+√11√(√29+√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