Geometric exponents, SLE and logarithmic minimal models

Saint-Aubin, Yvan, Pearce, Paul A. and Rasmussen, Jorgen (2009) Geometric exponents, SLE and logarithmic minimal models. Journal of Statistical Mechanics: Theory and Experiment, 2009 2: P02028.1-P02028.39. doi:10.1088/1742-5468/2009/02/P02028


Author Saint-Aubin, Yvan
Pearce, Paul A.
Rasmussen, Jorgen
Title Geometric exponents, SLE and logarithmic minimal models
Journal name Journal of Statistical Mechanics: Theory and Experiment   Check publisher's open access policy
ISSN 1742-5468
Publication date 2009-02
Sub-type Article (original research)
DOI 10.1088/1742-5468/2009/02/P02028
Volume 2009
Issue 2
Start page P02028.1
End page P02028.39
Total pages 39
Place of publication Bristol, United Kingdom
Publisher Institute of Physics Publishing
Language eng
Formatted abstract
In statistical mechanics, observables are usually related to local degrees of freedom such as the Q<4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal 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 or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to values of conformal dimensions not found among the finite sets of values allowed by the rational minimal models. Working in the context of a loop gas with loop fugacity β = -2cos(4π/κ), we use Monte Carlo simulations to measure the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the SLE parameter κ = (4p′/p) is rational with p<p′, we argue that the geometric exponents are related to conformal dimensions found in the infinitely extended Kac tables of the logarithmic minimal models . These theories describe lattice systems with non-local degrees of freedom. We present results 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 from SLEκ, with predictions from theoretical physics and with other numerical experiments. Throughout, we emphasize the relationships between SLEκ, geometric exponents and the conformal dimensions of the underlying CFTs.
Keyword Conformal field theory
Solvable lattice models
Classical Monte Carlo simulations
Critical exponents and amplitudes (theory)
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
 
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Created: Wed, 14 Mar 2012, 12:29:45 EST by Kay Mackie on behalf of Mathematics