315 research outputs found
Boundary states for a free boson defined on finite geometries
Langlands recently constructed a map that factorizes the partition function
of a free boson on a cylinder with boundary condition given by two arbitrary
functions in the form of a scalar product of boundary states. We rewrite these
boundary states in a compact form, getting rid of technical assumptions
necessary in his construction. This simpler form allows us to show explicitly
that the map between boundary conditions and states commutes with conformal
transformations preserving the boundary and the reality condition on the scalar
field.Comment: 16 pages, LaTeX (uses AMS components). Revised version; an analogy
with string theory computations is discussed and references adde
Critical Percolation in Finite Geometries
The methods of conformal field theory are used to compute the crossing
probabilities between segments of the boundary of a compact two-dimensional
region at the percolation threshold. These probabilities are shown to be
invariant not only under changes of scale, but also under mappings of the
region which are conformal in the interior and continuous on the boundary. This
is a larger invariance than that expected for generic critical systems.
Specific predictions are presented for the crossing probability between
opposite sides of a rectangle, and are compared with recent numerical work. The
agreement is excellent.Comment: 10 page
The Tails of the Crossing Probability
The scaling of the tails of the probability of a system to percolate only in
the horizontal direction was investigated numerically for correlated
site-bond percolation model for .We have to demonstrate that the
tails of the crossing probability far from the critical point have shape
where is the correlation
length index, is the probability of a bond to be closed. At
criticality we observe crossover to another scaling . Here is a scaling index describing the
central part of the crossing probability.Comment: 20 pages, 7 figures, v3:one fitting procedure is changed, grammatical
change
Statistical properties of the low-temperature conductance peak-heights for Corbino discs in the quantum Hall regime
A recent theory has provided a possible explanation for the ``non-universal
scaling'' of the low-temperature conductance (and conductivity) peak-heights of
two-dimensional electron systems in the integer and fractional quantum Hall
regimes. This explanation is based on the hypothesis that samples which show
this behavior contain density inhomogeneities. Theory then relates the
non-universal conductance peak-heights to the ``number of alternating
percolation clusters'' of a continuum percolation model defined on the
spatially-varying local carrier density. We discuss the statistical properties
of the number of alternating percolation clusters for Corbino disc samples
characterized by random density fluctuations which have a correlation length
small compared to the sample size. This allows a determination of the
statistical properties of the low-temperature conductance peak-heights of such
samples. We focus on a range of filling fraction at the center of the plateau
transition for which the percolation model may be considered to be critical. We
appeal to conformal invariance of critical percolation and argue that the
properties of interest are directly related to the corresponding quantities
calculated numerically for bond-percolation on a cylinder. Our results allow a
lower bound to be placed on the non-universal conductance peak-heights, and we
compare these results with recent experimental measurements.Comment: 7 pages, 4 postscript figures included. Revtex with epsf.tex and
multicol.sty. The revised version contains some additional discussion of the
theory and slightly improved numerical result
Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation
Extensive Monte-Carlo simulations were performed to evaluate the excess
number of clusters and the crossing probability function for three-dimensional
percolation on the simple cubic (s.c.), face-centered cubic (f.c.c.), and
body-centered cubic (b.c.c.) lattices. Systems L x L x L' with L' >> L were
studied for both bond (s.c., f.c.c., b.c.c.) and site (f.c.c.) percolation. The
excess number of clusters per unit length was confirmed to be a
universal quantity with a value . Likewise, the
critical crossing probability in the L' direction, with periodic boundary
conditions in the L x L plane, was found to follow a universal exponential
decay as a function of r = L'/L for large r. Simulations were also carried out
to find new precise values of the critical thresholds for site percolation on
the f.c.c. and b.c.c. lattices, yielding , .Comment: 14 pages, 7 figures, LaTeX, submitted to J. Phys. A: Math. Gen, added
references, corrected typo
Universal crossing probability in anisotropic systems
Scale-invariant universal crossing probabilities are studied for critical
anisotropic systems in two dimensions. For weakly anisotropic standard
percolation in a rectangular-shaped system, Cardy's exact formula is
generalized using a length-rescaling procedure. For strongly anisotropic
systems in 1+1 dimensions, exact results are obtained for the random walk with
absorbing boundary conditions, which can be considered as a linearized
mean-field approximation for directed percolation. The bond and site directed
percolation problem is itself studied numerically via Monte Carlo simulations
on the diagonal square lattice with either free or periodic boundary
conditions. A scale-invariant critical crossing probability is still obtained,
which is a universal function of the effective aspect ratio r_eff=c r where
r=L/t^z, z is the dynamical exponent and c is a non-universal amplitude.Comment: 7 pages, 4 figure
Conformal loop ensembles and the stress-energy tensor
We give a construction of the stress-energy tensor of conformal field theory
(CFT) as a local "object" in conformal loop ensembles CLE_\kappa, for all
values of \kappa in the dilute regime 8/3 < \kappa <= 4 (corresponding to the
central charges 0 < c <= 1, and including all CFT minimal models). We provide a
quick introduction to CLE, a mathematical theory for random loops in simply
connected domains with properties of conformal invariance, developed by
Sheffield and Werner (2006). We consider its extension to more general regions
of definition, and make various hypotheses that are needed for our construction
and expected to hold for CLE in the dilute regime. Using this, we identify the
stress-energy tensor in the context of CLE. This is done by deriving its
associated conformal Ward identities for single insertions in CLE probability
functions, along with the appropriate boundary conditions on simply connected
domains; its properties under conformal maps, involving the Schwarzian
derivative; and its one-point average in terms of the "relative partition
function." Part of the construction is in the same spirit as, but widely
generalizes, that found in the context of SLE_{8/3} by the author, Riva and
Cardy (2006), which only dealt with the case of zero central charge in simply
connected hyperbolic regions. We do not use the explicit construction of the
CLE probability measure, but only its defining and expected general properties.Comment: 49 pages, 3 figures. This is a concatenated, reduced and simplified
version of arXiv:0903.0372 and (especially) arXiv:0908.151
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