12 research outputs found
High-temperature expansion for Ising models on quasiperiodic tilings
We consider high-temperature expansions for the free energy of zero-field
Ising models on planar quasiperiodic graphs. For the Penrose and the octagonal
Ammann-Beenker tiling, we compute the expansion coefficients up to 18th order.
As a by-product, we obtain exact vertex-averaged numbers of self-avoiding
polygons on these quasiperiodic graphs. In addition, we analyze periodic
approximants by computing the partition function via the Kac-Ward determinant.
For the critical properties, we find complete agreement with the commonly
accepted conjecture that the models under consideration belong to the same
universality class as those on periodic two-dimensional lattices.Comment: 24 pages, 8 figures (EPS), uses IOP styles (included
Partition function zeros of aperiodic Ising models
We consider Ising models defined on periodic approximants of aperiodic
graphs. The model contains only a single coupling constant and no magnetic
field, so the aperiodicity is entirely given by the different local
environments of neighbours in the aperiodic graph. In this case, the partition
function zeros in the temperature variable, also known as the Fisher zeros, can
be calculated by diagonalisation of finite matrices. We present the partition
function zero patterns for periodic approximants of the Penrose and the
Ammann-Beenker tiling, and derive precise estimates of the critical
temperatures.Comment: Invited talk at QTS2, Krakow, July 2001; 6 pages, several postscript
figures, World Scientific proceedings LaTeX styl
Finite-lattice expansion for Ising models on quasiperiodic tilings
Low-temperature series are calculated for the free energy, magnetisation,
susceptibility and field-derivatives of the susceptibility in the Ising model
on the quasiperiodic Penrose lattice. The series are computed to order 20 and
estimates of the critical exponents alpha, beta and gamma are obtained from
Pade approximants.Comment: 16 pages, REVTeX, 26 postscript figure
Theoretical investigations of magnetic and electronic properties of quasicrystals
Es werden physikallische Eigenschaften von Quasikristallen
anhand von quasiperiodischen Ising- und Tight-Binding-Modellen
auf dem fuenfzaehligen Penrose- und achtzaehligen
Amman-Beenker-Muster untersucht.
Bei den Ising-Modellen wird eine graphische Hochtemperaturentwicklung
der freien Energie ausgerechnet und die kritischen Parameter
des ferromagnetischen Phasenueberganges abgeschaetzt. Weiterhin
wird mittels eines analytischen Resultates die freie Energie
auf den periodischen Approximanten quasiperiodischer Muster exakt ausgerechnet
und zur Bestimmung der Verteilung komplexer (Fisher-)Nullstellen
herangezogen. Letztendlich wird noch ein Ising-Modell mit einem verschiedenen,
nicht-Onsager kritischen Verhalten konstruiert und untersucht.
Im zweiten Kapitel werden kritische, nichtnormierbare
Eigenzustaende eines quasiperiodischen Tight-Binding-Modells
exakt berechnet. Es stellt sich heraus, dass die Eigenzustaende
eine selbstaehnliche, fraktale Struktur aufweisen
die in Details untersucht wird
Modeling of waiting times and price changes in currency exchange data
A theory which describes the share price evolution at financial markets as a continuous-time random walk has been generalized in order to take into account the dependence of waiting times t on price returns x. A joint probability density function (pdf) which uses the concept of a L\'{e}vy stable distribution is worked out. The theory is fitted to high-frequency US$/Japanese Yen exchange rate and low-frequency 19th century Irish stock data. The theory has been fitted both to price return and to waiting time data and the adherence to data, in terms of the chi-squared test statistic, has been improved when compared to the old theory.
Dynamics of Money and Income Distributions
We study the model of interacting agents proposed by Chatterjee et al that allows agents to both save and exchange wealth. Closed equations for the wealth distribution are developed using a mean field approximation. We show that when all agents have the same fixed savings propensity, subject to certain well defined approximations defined in the text, these equations yield the conjecture proposed by Chatterjee for the form of the stationary agent wealth distribution. If the savings propensity for the equations is chosen according to some random distribution we show further that the wealth distribution for large values of wealth displays a Pareto like power law tail, ie P(w)\sim w^{1+a}. However the value of for the model is exactly 1. Exact numerical simulations for the model illustrate how, as the savings distribution function narrows to zero, the wealth distribution changes from a Pareto form to to an exponential function. Intermediate regions of wealth may be approximately described by a power law with . However the value never reaches values of \~ 1.6-1.7 that characterise empirical wealth data. This conclusion is not changed if three body agent exchange processes are allowed. We conclude that other mechanisms are required if the model is to agree with empirical wealth data.