Numerical tests of schramm-loewner evolution in random lattice spin models

Abstract

Interfaces induced by symmetry-breaking boundary conditions in lattice spin models are possible realisations of SLE. For homogeneous systems with short-ranged interactions, it can be shown, even rigorously in some cases, that the assumptions behind SLE (Domain Markov property and conformal invariance) hold true. However, this is not the case for random systems. In recent years, numerical simulations gave some evidences of a possible description of interfaces in random systems by SLE. Here, different numerical tests will be reviewed, both of the basic assumptions which underlie SLE and of several derived consequences of the SLE description of critical phenomena, which have been tried up to now. A detailed exposition of these evidences in the cases of the random Potts model, the Solid-On-Solid model on a random substrate and the 2D Ising spin glass will be provided. © 2012 Springer-Verlag Berlin Heidelberg.