The geometry of fractal percolation

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: {itemize} the statements work for all directions, not almost all, the statements are true for more general projections, for example radial projections onto a circle, in the case $\dim_H >1$, each projection has not only positive Lebesgue measure but also has nonempty interior. {itemize}Comment: Survey submitted for AFRT2012 conferenc

Anisotropic Scaling in Layered Aperiodic Ising Systems

The influence of a layered aperiodic modulation of the couplings on the critical behaviour of the two-dimensional Ising model is studied in the case of marginal perturbations. The aperiodicity is found to induce anisotropic scaling. The anisotropy exponent z, given by the sum of the surface magnetization scaling dimensions, depends continuously on the modulation amplitude. Thus these systems are scale invariant but not conformally invariant at the critical point.Comment: 7 pages, 2 eps-figures, Plain TeX and epsf, minor correction

Surface Magnetization of Aperiodic Ising Systems: a Comparative Study of the Bond and Site Problems

We investigate the influence of aperiodic perturbations on the critical behaviour at a second order phase transition. The bond and site problems are compared for layered systems and aperiodic sequences generated through substitution. In the bond problem, the interactions between the layers are distributed according to an aperiodic sequence whereas in the site problem, the layers themselves follow the sequence. A relevance-irrelevance criterion introduced by Luck for the bond problem is extended to discuss the site problem. It involves a wandering exponent for pairs, which can be larger than the one considered before in the bond problem. The surface magnetization of the layered two-dimensional Ising model is obtained, in the extreme anisotropic limit, for the period-doubling and Thue-Morse sequences.Comment: 19 pages, Plain TeX, IOP macros + epsf, 6 postscript figures, minor correction

Dimension (in)equalities and H\"older continuous curves in fractal percolation

We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is a.s. the union of non-trivial H\"older continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.Comment: 22 pages, 3 figures, accepted for publication in Journal of Theoretical Probabilit

Radial Fredholm perturbation in the two-dimensional Ising model and gap-exponent relation

We consider concentric circular defects in the two-dimensional Ising model, which are distributed according to a generalized Fredholm sequence, i. e. at exponentially increasing radii. This type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation. The critical exponent of the local magnetization is obtained through finite-size scaling, using a corner transfer matrix approach in the extreme anisotropic limit. It varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial Hilhorst-van Leeuwen model. Through a conformal mapping of the system onto a strip, the gap-exponent relation is shown to remain valid for such an aperiodic defect.Comment: 12 pages, TeX file + 4 figures, epsf neede

Local critical behaviour at aperiodic surface extended perturbation in the Ising quantum chain

The surface critical behaviour of the semi--infinite one--dimensional quantum Ising model in a transverse field is studied in the presence of an aperiodic surface extended modulation. The perturbed couplings are distributed according to a generalized Fredholm sequence, leading to a marginal perturbation and varying surface exponents. The surface magnetic exponents are calculated exactly whereas the expression of the surface energy density exponent is conjectured from a finite--size scaling study. The system displays surface order at the bulk critical point, above a critical value of the modulation amplitude. It may be considered as a discrete realization of the Hilhorst--van Leeuwen model.Comment: 13 pages, TeX file + 6 figures, epsf neede

Comparison of the ICare® rebound tonometer with the Goldmann tonometer in a normal population

The aim of this study was to evaluate the accuracy of measurement of intraocular pressure (IOP) using a new induction/impact rebound tonometer (ICare) in comparison with the Goldmann applanation tonometer (AT). The left eyes of 46 university students were assessed with the two tonometers, with induction tonometry being performed first. The ICare was handled by an optometrist and the Goldmann tonometer by an ophthalmologist. In this study, statistically significant differences were found when comparing the ICare rebound tonometer with applanation tonometry (AT) (p < 0.05). The mean difference between the two tonometers was 1.34 +/- 2.03 mmHg (mean +/- S.D.) and the 95% limits of agreement were +/-3.98 mmHg. A frequency distribution of the differences demonstrated that in more than 80% of cases the IOP readings differed by <3 mmHg between the ICare and the AT. In the present population the ICare overestimates the IOP value by 1.34 mmHg on average when compared with Goldmann tonometer. Nevertheless, the ICare tonometer may be helpful as a screening tool when Goldmann applanation tonometry is not applicable or not recommended, as it is able to estimate IOP within a range of +/-3.00 mmHg in more than 80% of the populatio

Multidimensional Gaussian sums arising from distribution of Birkhoff sums in zero entropy dynamical systems

A duality formula, of the Hardy and Littlewood type for multidimensional Gaussian sums, is proved in order to estimate the asymptotic long time behavior of distribution of Birkhoff sums $S_n$ of a sequence generated by a skew product dynamical system on the $\mathbb{T}^2$ torus, with zero Lyapounov exponents. The sequence, taking the values $\pm 1$, is pairwise independent (but not independent) ergodic sequence with infinite range dependence. The model corresponds to the motion of a particle on an infinite cylinder, hopping backward and forward along its axis, with a transversal acceleration parameter $\alpha$. We show that when the parameter $\alpha /\pi$ is rational then all the moments of the normalized sums $E((S_n/\sqrt{n})^k)$, but the second, are unbounded with respect to n, while for irrational $\alpha /\pi$, with bounded continuous fraction representation, all these moments are finite and bounded with respect to n.Comment: To be published in J. Phys.

Common trends in the critical behavior of the Ising and directed walk models

We consider layered two-dimensional Ising and directed walk models and show that the two problems are inherently related. The information about the zero-field thermodynamical properties of the Ising model is contained into the transfer matrix of the directed walk. For several hierarchical and aperiodic distributions of the couplings, critical exponents for the two problems are obtained exactly through renormalization.Comment: 4 pages, RevTeX file + 1 figure, epsf needed. To be published in PR