263 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
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
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
Exact results at the 2-D percolation point
We derive exact expressions for the excess number of clusters b and the
excess cumulants b_n of a related quantity at the 2-D percolation point.
High-accuracy computer simulations are in accord with our predictions. b is a
finite-size correction to the Temperley-Lieb or Baxter-Temperley-Ashley formula
for the number of clusters per site n_c in the infinite system limit; the bn
correct bulk cumulants. b and b_n are universal, and thus depend only on the
system's shape. Higher-order corrections show no apparent dependence on
fractional powers of the system size.Comment: 12 pages, 2 figures, LaTeX, submitted to Physical Review Letter
Dirac cohomology, elliptic representations and endoscopy
The first part (Sections 1-6) of this paper is a survey of some of the recent
developments in the theory of Dirac cohomology, especially the relationship of
Dirac cohomology with (g,K)-cohomology and nilpotent Lie algebra cohomology;
the second part (Sections 7-12) is devoted to understanding the unitary
elliptic representations and endoscopic transfer by using the techniques in
Dirac cohomology. A few problems and conjectures are proposed for further
investigations.Comment: This paper will appear in `Representations of Reductive Groups, in
Honor of 60th Birthday of David Vogan', edited by M. Nervins and P. Trapa,
published by Springe
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