2d quantum gravity: loop nestings and CLE

We study random triangulations of the disk with V vertices, carrying a configuration of self-avoiding loops, each of them counted with a weight n. We answer the question: how is distributed the number (= nesting) of loops separating a randomly chosen point from the boundary of the disk ? and study the behavior of this distribution in the continuum limit (large V).

Physicists tell us that the (fractal) geometry of these triangulations with loops is supposedly described by the coupling of Liouville field theory with the so-called conformal loop ensembles (CLE_{k}) with n = -2\cos(4\pi/k) ; proving this identification is an important open problem in mathematical physics.  The CLE_{k} is a process of non-crossing loop configurations in the unit disk, whose distribution is conformally invariant, defined by Sheffield and Werner. Miller, Watson and Wilson proved a formula for the nesting multifractal spectral, i.e. the Hausdorff dimension of the set of points having a given value of nesting.

We show that our distribution of nestings (found for loops on random triangulations) reproduces the multifractal spectra of nestings in CLE after a suitable KPZ transformation that appear in the coupling to Liouville theory. This provides new evidence for the above mentioned conjecture. Our results also leads to prediction of new formulas for the multifractal spectral of CLE on the Riemann sphere.

This is based on joint work with Jeremie Bouttier and Bertrand Duplantier.