486 research outputs found
Two-Dimensional Copolymers and Exact Conformal Multifractality
We consider in two dimensions the most general star-shaped copolymer, mixing
random (RW) or self-avoiding walks (SAW) with specific interactions thereof.
Its exact bulk or boundary conformal scaling dimensions in the plane are all
derived from an algebraic structure existing on a random lattice (2D quantum
gravity). The multifractal dimensions of the harmonic measure of a 2D RW or SAW
are conformal dimensions of certain star copolymers, here calculated exactly as
non rational algebraic numbers. The associated multifractal function f(alpha)
are found to be identical for a random walk or a SAW in 2D. These are the first
examples of exact conformal multifractality in two dimensions.Comment: 4 pages, 2 figures, revtex, to appear in Phys. Rev. Lett., January
199
Conformally Invariant Fractals and Potential Theory
The multifractal (MF) distribution of the electrostatic potential near any
conformally invariant fractal boundary, like a critical O(N) loop or a
-state Potts cluster, is solved in two dimensions. The dimension of the boundary set with local wedge angle is , with the central charge of the
model. As a corollary, the dimensions
of the external perimeter and of the hull of a Potts cluster obey
the duality equation . A related covariant
MF spectrum is obtained for self-avoiding walks anchored at cluster boundaries.Comment: 5 pages, 1 figur
Duality and KPZ in Liouville Quantum Gravity
We present a (mathematically rigorous) probabilistic and geometrical proof of
the KPZ relation between scaling exponents in a Euclidean planar domain D and
in Liouville quantum gravity. It uses the properly regularized quantum area
measure d\mu_\gamma=\epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz
is Lebesgue measure on D, \gamma is a real parameter, 0\leq \gamma <2, and
h_\epsilon(z) denotes the mean value on the circle of radius \epsilon centered
at z of an instance h of the Gaussian free field on D. The proof extends to the
boundary geometry. The singular case \gamma >2 is shown to be related to the
quantum measure d\mu_{\gamma'}, \gamma' < 2, by the fundamental duality
\gamma\gamma'=4.Comment: 4 pages, 1 figur
Harmonic Measure and Winding of Conformally Invariant Curves
The exact joint multifractal distribution for the scaling and winding of the
electrostatic potential lines near any conformally invariant scaling curve is
derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff
dimension of the points where the potential scales with distance as while the curve logarithmically spirals with a rotation angle
phi=lambda ln r. It obeys the scaling law f(\alpha,\lambda)=(1+\lambda^2)
f(\bar \alpha)-b\lambda^2 with \bar \alpha=\alpha/(1+\lambda^2) and
b=(25-c)/{12}$, and where f(\alpha)\equiv f(\alpha,0) is the pure harmonic
measure spectrum, and c the conformal central charge. The results apply to O(N)
and Potts models, as well as to {\rm SLE}_{\kappa}.Comment: 3 figure
Intra-chain correlation functions and shapes of homopolymers with different architectures in dilute solution
We present results of Monte Carlo study of the monomer-monomer correlation
functions, static structure factor and asphericity characteristics of a single
homopolymer in the coil and globular states for three distinct architectures of
the chain: ring, open and star. To rationalise the results we introduce the
dimensionless correlation functions rescaled via the corresponding mean-squared
distances between monomers. For flexible chains with some architectures these
functions exhibit a large degree of universality by falling onto a single or
several distinct master curves. In the repulsive regime, where a stretched
exponential times a power law form (de Cloizeaux scaling) can be applied, the
corresponding exponents and have been obtained. The exponent
is found to be universal for flexible strongly repulsive coils
and in agreement with the theoretical prediction from improved higher-order
Borel-resummed renormalisation group calculations. The short-distance exponents
of an open flexible chain are in a good agreement with the
theoretical predictions in the strongly repulsive regime also. However,
increasing the Kuhn length in relation to the monomer size leads to their fast
cross-over towards the Gaussian behaviour. Likewise, a strong sensitivity of
various exponents on the stiffness of the chain, or on the number
of arms in star polymers, is observed. The correlation functions in the
globular state are found to have a more complicated oscillating behaviour and
their degree of universality has been reviewed. Average shapes of the polymers
in terms of the asphericity characteristics, as well as the universal behaviour
in the static structure factors, have been also investigated.Comment: RevTeX 12 pages, 10 PS figures. Accepted by J. Chem. Phy
Renormalization and Hyperscaling for Self-Avoiding Manifold Models
The renormalizability of the self-avoiding manifold (SAM) Edwards model is
established. We use a new short distance multilocal operator product expansion
(MOPE), which extends methods of local field theories to a large class of
models with non-local singular interactions. This validates the direct
renormalization method introduced before, as well as scaling laws. A new
general hyperscaling relation for the configuration exponent gamma is derived.
Manifolds at the Theta-point, and long range Coulomb interactions are briefly
discussed.Comment: 10 pages + 1 figure, TeX + harvmac & epsf (uuencoded file),
SPhT/93-07
Exact Multifractal Exponents for Two-Dimensional Percolation
The harmonic measure (or diffusion field or electrostatic potential) near a
percolation cluster in two dimensions is considered. Its moments, summed over
the accessible external hull, exhibit a multifractal spectrum, which I
calculate exactly. The generalized dimensions D(n) as well as the MF function
f(alpha) are derived from generalized conformal invariance, and are shown to be
identical to those of the harmonic measure on 2D random walks or self-avoiding
walks. An exact application to the anomalous impedance of a rough percolative
electrode is given. The numerical checks are excellent. Another set of exact
and universal multifractal exponents is obtained for n independent
self-avoiding walks anchored at the boundary of a percolation cluster. These
exponents describe the multifractal scaling behavior of the average nth moment
of the probabity for a SAW to escape from the random fractal boundary of a
percolation cluster in two dimensions.Comment: 5 pages, 3 figures (in colors
Two-dimensional Copolymers and Multifractality: Comparing Perturbative Expansions, MC Simulations, and Exact Results
We analyze the scaling laws for a set of two different species of long
flexible polymer chains joined together at one of their extremities (copolymer
stars) in space dimension D=2. We use a formerly constructed field-theoretic
description and compare our perturbative results for the scaling exponents with
recent conjectures for exact conformal scaling dimensions derived by a
conformal invariance technique in the context of D=2 quantum gravity. A simple
MC simulation brings about reasonable agreement with both approaches. We
analyse the remarkable multifractal properties of the spectrum of scaling
exponents.Comment: 5 page
Path Crossing Exponents and the External Perimeter in 2D Percolation
2D Percolation path exponents describe probabilities for
traversals of annuli by non-overlapping paths, each on either occupied
or vacant clusters, with at least one of each type. We relate the probabilities
rigorously to amplitudes of models whose exponents, believed to be
exact, yield . This extends to half-integers
the Saleur--Duplantier exponents for clusters, yields the exact
fractal dimension of the external cluster perimeter, , and also explains the absence of narrow gate fjords, as originally
found by Grossman and Aharony.Comment: 4 pages, 2 figures (EPSF). Revised presentatio
Renormalization of Crumpled Manifolds
We consider a model of D-dimensional tethered manifold interacting by
excluded volume in R^d with a single point. By use of intrinsic distance
geometry, we first provide a rigorous definition of the analytic continuation
of its perturbative expansion for arbitrary D, 0 < D < 2. We then construct
explicitly a renormalization operation, ensuring renormalizability to all
orders. This is the first example of mathematical construction and
renormalization for an interacting extended object with continuous internal
dimension, encompassing field theory.Comment: 10 pages (1 figure, included), harvmac, SPhT/92-15
- …