In this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm’s SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally-invariant scaling limit seems sufficient to deduce the required condition.
Our paper serves as an important step in establishing the convergence of Ising and FK Ising interfaces to SLE curves; moreover, the setup is adapted to branching interface trees, conjecturally describing the full interface picture by a collection of branching SLEs.
"Random curves, scaling limits and Loewner evolutions." Ann. Probab. 45 (2) 698 - 779, March 2017. https://doi.org/10.1214/15-AOP1074