From fully-packed loops to meanders: Exact exponents

We address the meander problem “enumerate all topologically inequivalent configurations of a closed nonselfintersecting plane curve intersecting a given line through a fixed number of points”. We show that meanders may be viewed as the configurations of a suitable fully-packed loop statistical model defined on a random surface. Using standard results relating critical singularities of a lattice model to its gravitational version on random surfaces, we predict the meander configuration exponent α = (29 + 145)/12 and many other meandric exponents.