Geometric exponents, SLE and logarithmic minimal models

Abstract

In statistical mechanics, observables are usually related to local degrees of freedom such as theQ<4 distinctstates of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuumscaling limit, these models are described by rational conformal field theories, namely theminimal models for suitable p,p′. More generally, as in stochastic Loewner evolution(SLEκ), one can consider observables related to non-local degrees of freedom such as paths orboundaries of clusters. This leads to fractal dimensions or geometric exponents related tovalues of conformal dimensions not found among the finite sets of values allowed by therational minimal models. Working in the context of a loop gas with loop fugacityβ = −2cos(4π/κ), we use Monte Carlo simulations to measure the fractal dimensions of variousgeometric objects such as paths and the generalizations of cluster mass, cluster hull,external perimeter and red bonds. Specializing to the case where the SLE parameterκ = (4p′/p) is rationalwith p<p′, we argue that the geometric exponents are related to conformal dimensions found in theinfinitely extended Kac tables of the logarithmic minimal models . These theories describe lattice systems with non-local degrees of freedom. We presentresults for critical dense polymers , critical percolation , the logarithmic Ising model , the logarithmic tricritical Ising model as well as . Our results are compared with rigorous results fromSLEκ, with predictions from theoretical physics and with other numericalexperiments. Throughout, we emphasize the relationships betweenSLEκ, geometric exponents and the conformal dimensions of the underlying CFTs.