1,475 research outputs found
Critical Exponents near a Random Fractal Boundary
The critical behaviour of correlation functions near a boundary is modified
from that in the bulk. When the boundary is smooth this is known to be
characterised by the surface scaling dimension \xt. We consider the case when
the boundary is a random fractal, specifically a self-avoiding walk or the
frontier of a Brownian walk, in two dimensions, and show that the boundary
scaling behaviour of the correlation function is characterised by a set of
multifractal boundary exponents, given exactly by conformal invariance
arguments to be \lambda_n = 1/48 (\sqrt{1+24n\xt}+11)(\sqrt{1+24n\xt}-1).
This result may be interpreted in terms of a scale-dependent distribution of
opening angles of the fractal boundary: on short distance scales these
are sharply peaked around . Similar arguments give the
multifractal exponents for the case of coupling to a quenched random bulk
geometry.Comment: 13 pages. Comments on relation to results in quenched random bulk
added, and on relation to other recent work. Typos correcte
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
Calogero-Sutherland eigenfunctions with mixed boundary conditions and conformal field theory correlators
We construct certain eigenfunctions of the Calogero-Sutherland hamiltonian
for particles on a circle, with mixed boundary conditions. That is, the
behavior of the eigenfunction, as neighbouring particles collide, depend on the
pair of colliding particles. This behavior is generically a linear combination
of two types of power laws, depending on the statistics of the particles
involved. For fixed ratio of each type at each pair of neighboring particles,
there is an eigenfunction, the ground state, with lowest energy, and there is a
discrete set of eigenstates and eigenvalues, the excited states and the
energies above this ground state. We find the ground state and special excited
states along with their energies in a certain class of mixed boundary
conditions, interpreted as having pairs of neighboring bosons and other
particles being fermions. These particular eigenfunctions are characterised by
the fact that they are in direct correspondence with correlation functions in
boundary conformal field theory. We expect that they have applications to
measures on certain configurations of curves in the statistical O(n) loop
model. The derivation, although completely independent from results of
conformal field theory, uses ideas from the "Coulomb gas" formulation.Comment: 35 pages, 9 figure
The Number of Incipient Spanning Clusters in Two-Dimensional Percolation
Using methods of conformal field theory, we conjecture an exact form for the
probability that n distinct clusters span a large rectangle or open cylinder of
aspect ratio k, in the limit when k is large.Comment: 9 pages, LaTeX, 1 eps figure. Additional references and comparison
with existing numerical results include
Critical behaviour in parabolic geometries
We study two-dimensional systems with boundary curves described by power
laws. Using conformal mappings we obtain the correlations at the bulk critical
point. Three different classes of behaviour are found and explained by scaling
arguments which also apply to higher dimensions. For an Ising system of
parabolic shape the behaviour of the order at the tip is also found.Comment: Old paper, for archiving. 6 pages, 1 figure, epsf, IOP macr
Effect of Random Impurities on Fluctuation-Driven First Order Transitions
We analyse the effect of quenched uncorrelated randomness coupling to the
local energy density of a model consisting of N coupled two-dimensional Ising
models. For N>2 the pure model exhibits a fluctuation-driven first order
transition, characterised by runaway renormalisation group behaviour. We show
that the addition of weak randomness acts to stabilise these flows, in such a
way that the trajectories ultimately flow back towards the pure decoupled Ising
fixed point, with the usual critical exponents alpha=0, nu=1, apart from
logarithmic corrections. We also show by examples that, in higher dimensions,
such transitions may either become continuous or remain first order in the
presence of randomness.Comment: 13 pp., LaTe
Mean Area of Self-Avoiding Loops
The mean area of two-dimensional unpressurised vesicles, or self-avoiding
loops of fixed length , behaves for large as , while their
mean square radius of gyration behaves as . The amplitude ratio
is computed exactly and found to equal . The physics of the
pressurised case, both in the inflated and collapsed phases, may be usefully
related to that of a complex O(n) field theory coupled to a U(1) gauge field,
in the limit .Comment: 12 pages, plain TeX, (one TeX macro omission corrected
Junctions of anyonic Luttinger wires
We present an extended study of anyonic Luttinger liquids wires jointing at a
single point. The model on the full line is solved with bosonization and the
junction of an arbitrary number of wires is treated imposing boundary
conditions that preserve exact solvability in the bosonic language. This allows
to reach, in the low momentum regime, some of the critical fixed points found
with the electronic boundary conditions. The stability of all the fixed points
is discussed.Comment: 16 pages, 2 figures, typos corrected, Refs adde
A Monte Carlo study of the triangular lattice gas with the first- and the second-neighbor exclusions
We formulate a Swendsen-Wang-like version of the geometric cluster algorithm.
As an application,we study the hard-core lattice gas on the triangular lattice
with the first- and the second-neighbor exclusions. The data are analyzed by
finite-size scaling, but the possible existence of logarithmic corrections is
not considered due to the limited data. We determine the critical chemical
potential as and the critical particle density as
. The thermal and magnetic exponents
and , estimated from Binder ratio and
susceptibility , strongly support the general belief that the model is in
the 4-state Potts universality class. On the other hand, the analyses of
energy-like quantities yield the thermal exponent ranging from
to . These values differ significantly from the expected value 3/2,
and thus imply the existence of logarithmic corrections.Comment: 4 figures 2 table
Universal Amplitude Combinations for Self-Avoiding Walks, Polygons and Trails
We give exact relations for a number of amplitude combinations that occur in
the study of self-avoiding walks, polygons and lattice trails. In particular,
we elucidate the lattice-dependent factors which occur in those combinations
which are otherwise universal, show how these are modified for oriented
lattices, and give new results for amplitude ratios involving even moments of
the area of polygons. We also survey numerical results for a wide range of
amplitudes on a number of oriented and regular lattices, and provide some new
ones.Comment: 20 pages, NI 92016, OUTP 92-54S, UCSBTH-92-5
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