Conformal Invariance of 2D Lattice Models

Specialeforsvar ved Ulrik Thinggaard Hansen

Titél: Conformal Invariance of 2D Lattice Models

Abstract: In this thesis, we give the seminal proof due to Smirnov et al that SLE16=3 arises as the scaling limit of the interface in the FK-Ising Model under Dobrushin Boundary Conditions. We introduce the four equivalent geometric and conformal conditions G2;G3;C2 and C3 for families of probability distributions on simple curves in C that guarantee that the probability distributions are pre-compact in the probability distributions on Loewner chains. Furthermore, we introduce several discrete analogues to classical complex analysis, and use these to show that FK-Ising interface satisfies condition G2.
Our main contribution is a unified framework for the proof of the theorem, which otherwise is spread over numerous articles, as well as the rigorous development of the theory of winding of a discrete curve. Furthermore, though somewhat unrelated, we give an original, self-contained proof that Caratheodory Convergence implies weak convergence of harmonic measure.

Vejleder: Bergfinnur Durhuus
Censor:   Carsten Lunde Petersen, RUC