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 , and the semi-meander exponent

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. 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.