On the direct SLE-martingale approach to the integrability of Conformal Loop Ensembles

Abstract

This paper is about the derivation of "integrability results" (i.e., for example, explicit formulas for the probability of events involving a few marked points) for Conformal Loop Ensembles, that are the (conjectured) scaling limits of discrete crititical planar lattice models from Statistical Physics. We discuss the explicit solutions to the PDEs provided by the original SLE/CLE martingale approach that AI now makes it possible to uncover. This provides short and direct new proofs of a number of formulas that have recently been derived based on Liouville quantum gravity, conformal field theory or Coulomb gas methods, and also makes it possible to derive some new ones. We briefly illustrate this with the examples of the backbone exponent and annulus crossing probabilities for critical percolation, but this direct approach seems to apply to a wide class of observables of two-dimensional systems related to conformal field theories.