Critical Percolation, Universality, and SLE6

Since the introduction of the Schramm-Loewner-Evolution (SLE) in 2000 ([25]), tremendous progress has been made in rigorously understanding the scaling limits of various 2D critical statistical mechanics models in two dimensions (see [22]). The starting point of understanding the scaling limit of a 2D critical lattice model is to consider the model on a bounded domain O ⊂ R2 and find a suitable observable at the discrete level which satisfies some discrete analyticity or harmonicity limit and, together with establishment of suitable boundary values, leads to conformal invariance in the continuum limit. For percolation, the appropriate observable is the crossing probability - conjectured to converge to the so-called Cardy's Formula in the continuum. In [7], Smirnov established conformal invariance of critical site percolation on the triangular lattice (in the scaling limit) by considering a triplet of observables related to crossing probability. However, Smirnov's proof takes advantage of the complete symmetry in the case of site percolation on the triangular lattice, and the triplet observables do not easily adapt themselves to percolation on other lattices.;This dissertation, representing joint work with L. Chayes and I. Binder (see [4], [5], [1], [2], [3]), contains construction of a non-trivial class of models for which we establish Cardy's Formula and, following the approach outlined in [8], establishes convergence to SLE6 for the law of the interface, thus establishing some limited statement of universality. In the course of (and in addition to) accomplishing this, we obtain some results which may find applicability to other percolation models: (1) We show how to extract Cardy's Formula given some interior analyticity statement (this requires some treatment of the discretization procedure in relation to retrieval of suitable boundary values) for a general class of domains; (2) our convergence to SLE6 proof is applicable for any percolation model satisfying reasonable assumptions and for which Cardy's Formula can be established; (3) we obtain some (almost) uniform estimates on crossing probabilities which may lead to some statement of rate of convergence to SLE6.