Critical behavior of the 3-state Potts model on Sierpinski carpet

We study the critical behavior of the 3-state Potts model, where the spins are located at the centers of the occupied squares of the deterministic Sierpinski carpet. A finite-size scaling analysis is performed from Monte Carlo simulations, for a Hausdorff dimension $d_{f}$ $\simeq 1.8928$. The phase transition is shown to be a second order one. The maxima of the susceptibility of the order parameter follow a power law in a very reliable way, which enables us to calculate the ratio of the exponents $\gamma /\nu$. We find that the scaling corrections affect the behavior of most of the thermodynamical quantities. However, the sequence of intersection points extracted from the Binder's cumulant provides bounds for the critical temperature. We are able to give the bounds for the exponent $1/\nu$ as well as for the ratio of the exponents $\beta/\nu$, which are compatible with the results calculated from the hyperscaling relation.Comment: 13 pages, 4 figure

Apparent Fractality Emerging from Models of Random Distributions

The fractal properties of models of randomly placed $n$-dimensional spheres ($n$=1,2,3) are studied using standard techniques for calculating fractal dimensions in empirical data (the box counting and Minkowski-sausage techniques). Using analytical and numerical calculations it is shown that in the regime of low volume fraction occupied by the spheres, apparent fractal behavior is observed for a range of scales between physically relevant cut-offs. The width of this range, typically spanning between one and two orders of magnitude, is in very good agreement with the typical range observed in experimental measurements of fractals. The dimensions are not universal and depend on density. These observations are applicable to spatial, temporal and spectral random structures. Polydispersivity in sphere radii and impenetrability of the spheres (resulting in short range correlations) are also introduced and are found to have little effect on the scaling properties. We thus propose that apparent fractal behavior observed experimentally over a limited range may often have its origin in underlying randomness.Comment: 19 pages, 12 figures. More info available at http://www.fh.huji.ac.il/~dani

Drip Paintings and Fractal Analysis

It has been claimed [1-6] that fractal analysis can be applied to unambiguously characterize works of art such as the drip paintings of Jackson Pollock. This academic issue has become of more general interest following the recent discovery of a cache of disputed Pollock paintings. We definitively demonstrate here, by analyzing paintings by Pollock and others, that fractal criteria provide no information about artistic authenticity. This work has also led to two new results in fractal analysis of more general scientific significance. First, the composite of two fractals is not generally scale invariant and exhibits complex multifractal scaling in the small distance asymptotic limit. Second the statistics of box-counting and related staircases provide a new way to characterize geometry and distinguish fractals from Euclidean objects

Escape from the vicinity of fractal basin boundaries of a star cluster

The dissolution process of star clusters is rather intricate for theory. We investigate it in the context of chaotic dynamics. We use the simple Plummer model for the gravitational field of a star cluster and treat the tidal field of the Galaxy within the tidal approximation. That is, a linear approximation of tidal forces from the Galaxy based on epicyclic theory in a rotating reference frame. The Poincar\'e surfaces of section reveal the effect of a Coriolis asymmetry. The system is non-hyperbolic which has important consequences for the dynamics. We calculated the basins of escape with respect to the Lagrangian points $L_1$ and $L_2$. The longest escape times have been measured for initial conditions in the vicinity of the fractal basin boundaries. Furthermore, we computed the chaotic saddle for the system and its stable and unstable manifolds. The chaotic saddle is a fractal structure in phase space which has the form of a Cantor set and introduces chaos into the system.Comment: Accepted by MNRAS, Figures have lower qualit

Fractal initial conditions and natural parameter values in hybrid inflation

We show that the initial field values required to produce inflation in the two fields original hybrid model, and its supergravity F-term extension, do not suffer from any fine-tuning problem, even when the fields are restricted to be sub-planckian and for almost all potential parameter values. This is due to the existence of an initial slow-roll violating evolution which has been overlooked so far. Due to the attractor nature of the inflationary valley, these trajectories end up producing enough accelerated expansion of the universe. By numerically solving the full non-linear dynamics, we show that the set of such successful initial field values is connected, of dimension two and possesses a fractal boundary of infinite length exploring the whole field space. We then perform a Monte-Carlo-Markov-Chain analysis of the whole parameter space consisting of the initial field values, field velocities and potential parameters. We give the marginalised posterior probability distributions for each of these quantities such that the universe inflates long enough to solve the usual cosmological problems. Inflation in the original hybrid model and its supergravity version appears to be generic and more probable by starting outside of the inflationary valley. Finally, the implication of our findings in the context of the eternal inflationary scenario are discussed.Comment: 16 pages, 16 figures, uses RevTeX. Lyapunov exponents and references added, matches published versio