3,189 research outputs found
The Loewner equation: maps and shapes
In the last few years, new insights have permitted unexpected progress in the
study of fractal shapes in two dimensions. A new approach, called
Schramm-Loewner evolution, or SLE, has arisen through analytic function theory
and probability theory, and given a new way of calculating fractal shapes in
critical phenomena, the theory of random walks, and of percolation. We present
a non-technical discussion of this development aimed to attract the attention
of condensed matter community to this fascinating subject
Approximation, generalisation and deconstruction of planar curves
CISRG discussion paper ; 1
Multicritical continuous random trees
We introduce generalizations of Aldous' Brownian Continuous Random Tree as
scaling limits for multicritical models of discrete trees. These discrete
models involve trees with fine-tuned vertex-dependent weights ensuring a k-th
root singularity in their generating function. The scaling limit involves
continuous trees with branching points of order up to k+1. We derive explicit
integral representations for the average profile of this k-th order
multicritical continuous random tree, as well as for its history distributions
measuring multi-point correlations. The latter distributions involve
non-positive universal weights at the branching points together with fractional
derivative couplings. We prove universality by rederiving the same results
within a purely continuous axiomatic approach based on the resolution of a set
of consistency relations for the multi-point correlations. The average profile
is shown to obey a fractional differential equation whose solution involves
hypergeometric functions and matches the integral formula of the discrete
approach.Comment: 34 pages, 12 figures, uses lanlmac, hyperbasics, eps
2D growth processes: SLE and Loewner chains
This review provides an introduction to two dimensional growth processes.
Although it covers a variety processes such as diffusion limited aggregation,
it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner
evolutions (SLE) which are Markov processes describing interfaces in 2D
critical systems. It starts with an informal discussion, using numerical
simulations, of various examples of 2D growth processes and their connections
with statistical mechanics. SLE is then introduced and Schramm's argument
mapping conformally invariant interfaces to SLE is explained. A substantial
part of the review is devoted to reveal the deep connections between
statistical mechanics and processes, and more specifically to the present
context, between 2D critical systems and SLE. Some of the SLE remarkable
properties are explained, as well as the tools for computing with SLE. This
review has been written with the aim of filling the gap between the
mathematical and the physical literatures on the subject.Comment: A review on Stochastic Loewner evolutions for Physics Reports, 172
pages, low quality figures, better quality figures upon request to the
authors, comments welcom
Scaling and Universality in City Space Syntax: between Zipf and Matthew
We report about universality of rank-integration distributions of open spaces
in city space syntax similar to the famous rank-size distributions of cities
(Zipf's law). We also demonstrate that the degree of choice an open space
represents for other spaces directly linked to it in a city follows a power law
statistic. Universal statistical behavior of space syntax measures uncovers the
universality of the city creation mechanism. We suggest that the observed
universality may help to establish the international definition of a city as a
specific land use pattern.Comment: 24 pages, 5 *.eps figure
Genetic Algorithm Optimization of a High-Directivity Microstrip Patch Antenna Having a Rectangular Profile
A single high-directivity microstrip patch antenna (MPA) having a rectangular profile, which can substitute a linear array is proposed. It is designed by using genetic algorithms with the advantage of not requiring a feeding network. The patch fits inside an area of 2.54λ x 0.25λ, resulting in a broadside pattern with a directivity of 12 dBi and a fractional impedance bandwidth of 4%. The antenna is fabricated and the measurements are in good agreement with the simulated results. The genetic MPA provides a similar directivity as linear arrays using a corporate or series feeding, with the advantage that the genetic MPA results in more bandwidth
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