178 research outputs found
General solution of an exact correlation function factorization in conformal field theory
We discuss a correlation function factorization, which relates a three-point
function to the square root of three two-point functions. This factorization is
known to hold for certain scaling operators at the two-dimensional percolation
point and in a few other cases. The correlation functions are evaluated in the
upper half-plane (or any conformally equivalent region) with operators at two
arbitrary points on the real axis, and a third arbitrary point on either the
real axis or in the interior. This type of result is of interest because it is
both exact and universal, relates higher-order correlation functions to
lower-order ones, and has a simple interpretation in terms of cluster or loop
probabilities in several statistical models. This motivated us to use the
techniques of conformal field theory to determine the general conditions for
its validity.
Here, we discover a correlation function which factorizes in this way for any
central charge c, generalizing previous results. In particular, the
factorization holds for either FK (Fortuin-Kasteleyn) or spin clusters in the
Q-state Potts models; it also applies to either the dense or dilute phases of
the O(n) loop models. Further, only one other non-trivial set of highest-weight
operators (in an irreducible Verma module) factorizes in this way. In this case
the operators have negative dimension (for c < 1) and do not seem to have a
physical realization.Comment: 7 pages, 1 figure, v2 minor revision
The density of critical percolation clusters touching the boundaries of strips and squares
We consider the density of two-dimensional critical percolation clusters,
constrained to touch one or both boundaries, in infinite strips, half-infinite
strips, and squares, as well as several related quantities for the infinite
strip. Our theoretical results follow from conformal field theory, and are
compared with high-precision numerical simulation. For example, we show that
the density of clusters touching both boundaries of an infinite strip of unit
width (i.e. crossing clusters) is proportional to .
We also determine numerically contours for the density of clusters crossing
squares and long rectangles with open boundaries on the sides, and compare with
theory for the density along an edge.Comment: 11 pages, 6 figures. Minor revision
Factorization of correlations in two-dimensional percolation on the plane and torus
Recently, Delfino and Viti have examined the factorization of the three-point
density correlation function P_3 at the percolation point in terms of the
two-point density correlation functions P_2. According to conformal invariance,
this factorization is exact on the infinite plane, such that the ratio R(z_1,
z_2, z_3) = P_3(z_1, z_2, z_3) [P_2(z_1, z_2) P_2(z_1, z_3) P_2(z_2,
z_3)]^{1/2} is not only universal but also a constant, independent of the z_i,
and in fact an operator product expansion (OPE) coefficient. Delfino and Viti
analytically calculate its value (1.022013...) for percolation, in agreement
with the numerical value 1.022 found previously in a study of R on the
conformally equivalent cylinder. In this paper we confirm the factorization on
the plane numerically using periodic lattices (tori) of very large size, which
locally approximate a plane. We also investigate the general behavior of R on
the torus, and find a minimum value of R approx. 1.0132 when the three points
are maximally separated. In addition, we present a simplified expression for R
on the plane as a function of the SLE parameter kappa.Comment: Small corrections (final version). In press, J. Phys.
Twist operator correlation functions in O(n) loop models
Using conformal field theoretic methods we calculate correlation functions of
geometric observables in the loop representation of the O(n) model at the
critical point. We focus on correlation functions containing twist operators,
combining these with anchored loops, boundaries with SLE processes and with
double SLE processes.
We focus further upon n=0, representing self-avoiding loops, which
corresponds to a logarithmic conformal field theory (LCFT) with c=0. In this
limit the twist operator plays the role of a zero weight indicator operator,
which we verify by comparison with known examples. Using the additional
conditions imposed by the twist operator null-states, we derive a new explicit
result for the probabilities that an SLE_{8/3} wind in various ways about two
points in the upper half plane, e.g. that the SLE passes to the left of both
points.
The collection of c=0 logarithmic CFT operators that we use deriving the
winding probabilities is novel, highlighting a potential incompatibility caused
by the presence of two distinct logarithmic partners to the stress tensor
within the theory. We provide evidence that both partners do appear in the
theory, one in the bulk and one on the boundary and that the incompatibility is
resolved by restrictive bulk-boundary fusion rules.Comment: 18 pages, 8 figure
Anchored Critical Percolation Clusters and 2-D Electrostatics
We consider the densities of clusters, at the percolation point of a
two-dimensional system, which are anchored in various ways to an edge. These
quantities are calculated by use of conformal field theory and computer
simulations. We find that they are given by simple functions of the potentials
of 2-D electrostatic dipoles, and that a kind of superposition {\it cum}
factorization applies. Our results broaden this connection, already known from
previous studies, and we present evidence that it is more generally valid. An
exact result similar to the Kirkwood superposition approximation emerges.Comment: 4 pages, 1 (color) figure. More numerics, minor corrections,
references adde
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