44 research outputs found
The Jordan Structure of Two Dimensional Loop Models
We show how to use the link representation of the transfer matrix of
loop models on the lattice to calculate partition functions, at criticality, of
the Fortuin-Kasteleyn model with various boundary conditions and parameter
and, more specifically,
partition functions of the corresponding -Potts spin models, with
. The braid limit of is shown to be a central element
of the Temperley-Lieb algebra , its eigenvalues are
determined and, for generic , a basis of its eigenvectors is constructed
using the Wenzl-Jones projector. To any element of this basis is associated a
number of defects , , and the basis vectors with the same
span a sector. Because components of these eigenvectors are singular when and , the link representations of
and are shown to have Jordan blocks between sectors and
when and (). When
and do not satisfy the previous constraint, is diagonalizable.Comment: 55 page
Geometric Exponents of Dilute Logarithmic Minimal Models
The fractal dimensions of the hull, the external perimeter and of the red
bonds are measured through Monte Carlo simulations for dilute minimal models,
and compared with predictions from conformal field theory and SLE methods. The
dilute models used are those first introduced by Nienhuis. Their loop fugacity
is beta = -2cos(pi/barkappa}) where the parameter barkappa is linked to their
description through conformal loop ensembles. It is also linked to conformal
field theories through their central charges c = 13 - 6(barkappa +
barkappa^{-1}) and, for the minimal models of interest here, barkappa = p/p'
where p and p' are two coprime integers. The geometric exponents of the hull
and external perimeter are studied for the pairs (p,p') = (1,1), (2,3), (3,4),
(4,5), (5,6), (5,7), and that of the red bonds for (p,p') = (3,4). Monte Carlo
upgrades are proposed for these models as well as several techniques to improve
their speeds. The measured fractal dimensions are obtained by extrapolation on
the lattice size H,V -> infinity. The extrapolating curves have large slopes;
despite these, the measured dimensions coincide with theoretical predictions up
to three or four digits. In some cases, the theoretical values lie slightly
outside the confidence intervals; explanations of these small discrepancies are
proposed.Comment: 41 pages, 32 figures, added reference
The representation theory of seam algebras
The boundary seam algebras were introduced by Morin-Duchesne,
Ridout and Rasmussen to formulate algebraically a large class of boundary conditions
for two-dimensional statistical loop models. The representation theory of these algebras is given: their irreducible, standard (cellular) and principal modules
are constructed and their structure explicited in terms of their composition factors and
of non-split short exact sequences. The dimensions of the irreducible modules and of the
radicals of standard ones are also given. The methods proposed here might be applicable
to a large family of algebras, for example to those introduced recently by Flores and
Peltola, and Crampé and Poulain d’Andecy