Harmonic measure and winding of random conformal paths: A Coulomb gas perspective

We consider random conformally invariant paths in the complex plane (SLEs). Using the Coulomb gas method in conformal field theory, we rederive the mixed multifractal exponents associated with both the harmonic measure and winding (rotation or monodromy) near such critical curves, previously obtained by quantum gravity methods. The results also extend to the general cases of harmonic measure moments and winding of multiple paths in a star configuration.     

Harmonic Measure and SLE

In this paper we study the multifractal structure of Schramm’s SLE curves. We derive the values of the (average) spectrum of harmonic measure and prove Duplantier’s prediction for the multifractal spectrum of SLE curves. The spectrum can also be used to derive estimates of the dimension, Hölder exponent and other geometrical quantities. The SLE curves provide perhaps the only example of sets where the spectrum is non-trivial yet exactly computable.     

Exact multifractal spectra for arbitrary laplacian random walks

Iterated conformal mappings are used to obtain exact multifractal spectra of the harmonic measure for arbitrary Laplacian random walks in two dimensions. Separate spectra are found to describe scaling of the growth measure in time, of the measure near the growth tip, and of the measure away from the growth tip. The spectra away from the tip coincide with those of conformally invariant equilibrium systems with arbitrary central charge c < or = 1, with c related to the particular walk chosen, while the scaling in time and near the tip cannot be obtained from the equilibrium properties.     

Harmonic measure and winding of conformally invariant curves

The exact joint multifractal distribution for the scaling and winding of the electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff dimension of the points where the potential scales with distance r as H approximately r(alpha) while the curve logarithmically spirals with a rotation angle phi=lambdalnr. It obeys the scaling law f(alpha,lambda)=(1+lambda(2))f(alpha)-blambda(2) with alpha=alpha/(1+lambda(2)) and b=(25-c)/12, and where f(alpha) identical with f(alpha,0) is the pure harmonic measure spectrum, and c the conformal central charge. The results apply to O(N) and Potts models, as well as to stochastic L?wner evolution.     

On the equality of Hausdorff and box counting dimensions

By viewing the covers of a fractal as a statistical mechanical system, the exact capacity of a multifractal is computed. The procedure can be extended to any multifractal described by a scaling function to show why the capacity and Hausdorff dimension are expected to be equal.     

Multifractal analysis of infinite products

We construct a family of measures called infinite products which generalize Gibbs measures in the one-dimensional lattice gas model. The multifractal properties of these measures are studied under some regularity conditions. In particular, if the -function is differentiable, we prove a formula which gives the Hausdorff dimension and packing dimension of the set of singularity points of a given order. Mathematical examples include Riesz products,g-measures, andG-measures.     

Harmonic measure for percolation and ising clusters including rare events

We obtain the harmonic measure of the hulls of critical percolation clusters and Ising-model Fortuin-Kastelyn clusters using a biased random-walk sampling technique which allows us to measure probabilities as small as 10{-300}. We find the multifractal D(q) spectrum including regions of small and negative q. Our results for external hulls agree with Duplantier's theoretical predictions for D(q) and his exponent -23/24 for the harmonic measure probability distribution for percolation. For the complete hull, we find the probability decays with an exponent of -1 for both systems.     

The fractal energy measurement and the singularity energy spectrum analysis

The singularity exponent (SE) is the characteristic parameter of fractal and multifractal signals. Based on SE, the fractal dimension reflecting the global self-similar character, the instantaneous SE reflecting the local self-similar character, the multifractal spectrum (MFS) reflecting the distribution of SE, and the time-varying MFS reflecting pointwise multifractal spectrum were proposed. However, all the studies were based on the depiction of spatial or differentiability characters of fractal signals. Taking the SE as the independent dimension, this paper investigates the fractal energy measurement (FEM) and the singularity energy spectrum (SES) theory. Firstly, we study the energy measurement and the energy spectrum of a fractal signal in the singularity domain, propose the conception of FEM and SES of multifractal signals, and investigate the Hausdorff measure and the local direction angle of the fractal energy element. Then, we prove the compatibility between FEM and traditional energy, and point out that SES can be measured in the fractal space. Finally, we study the algorithm of SES under the condition of a continuous signal and a discrete signal, and give the approximation algorithm of the latter, and the estimations of FEM and SES of the Gaussian white noise, Fractal Brownian motion and the multifractal Brownian motion show the theoretical significance and application value of FEM and SES.     

On Measures Driven by Markov Chains

We study measures on \([0,1]\) which are driven by a finite Markov chain and which generalize the famous Bernoulli products.We propose a hands-on approach to determine the structure function \(\tau \) and to prove that the multifractal formalism is satisfied. Formulas for the dimension of the measures and for the Hausdorff dimension of their supports are also provided. Finally, we identify the measures with maximal dimension.     

Harmonic measure of critical curves

Fractal geometry of critical curves appearing in 2D critical systems is characterized by their harmonic measure. For systems described by conformal field theories with central charge c < or = 1, scaling exponents of the harmonic measure have been computed by Duplantier [Phys. Rev. Lett. 84, 1363 (2000)10.1103/PhysRevLett.84.1363] by relating the problem to boundary two-dimensional gravity. We present a simple argument connecting the harmonic measure of critical curves to operators obtained by fusion of primary fields and compute characteristics of the fractal geometry by means of regular methods of conformal field theory. The method is not limited to theories with c < or = 1.     

Long-term correlations and multifractal nature in the intertrade durations of a liquid Chinese stock and its warrant

The intertrade duration of equities is an important financial measure, characterizing trading activities; it is defined as the waiting time between successive trades of an equity. Using the ultrahigh-frequency data of a liquid Chinese stock and its associated warrant, we perform a comparative investigation of the statistical properties of their intertrade duration time series. The distributions of the two equities can be better described by the shifted power-law form than the Weibull form, and their scaled distributions do not collapse onto a single curve. Although the intertrade durations of the two equities have very different magnitude, their intraday patterns exhibit very similar shapes. Both detrended fluctuation analysis (DFA) and detrending moving average analysis (DMA) show that the 1 min intertrade duration time series of the two equities are strongly correlated. In addition, both multifractal detrended fluctuation analysis (MFDFA) and multifractal detrending moving average analysis (MFDMA) unveil that the 1 min intertrade durations possess multifractal nature. However, the difference between the two singularity spectra of the two equities obtained from the MFDMA is much smaller than that from the MFDFA.     

On the multifractal analysis of Bernoulli convolutions. II. Dimensions

We show how the formalism developed in a previous paper allows us to exhibit the multifractal nature of the infinitely convolved Bernoulli measures for the golden mean. In this second part we show how the Hausdorff dimension of the set where the measure has a power law singularity of strength is related to the large-deviation function given in Part I.     

Measure-theoretical properties of the unstable foliation of two-dimensional differentiable area-preserving systems

This article analyzes in detail the statistical and measure-theoretical properties of the nonuniform stationary measure, referred to as the w-invariant measure, associated with the spatial length distribution of the integral manifolds of the unstable invariant foliation in two-dimensional differentiable area-preserving systems. The analysis is developed starting from a sequence of analytical approximations for the associated density. These approximations are related to the properties of the Jacobian matrix of the nth iteration of a Poincaré map. The w-invariant measure plays a fundamental role in the study of transport phenomena in laminar-chaotic fluid-mixing systems, for which it furnishes the asymptotic invariant distribution of intermaterial contact length between two fluids. The w-invariant measure turns out to be singular and exhibits multifractal features. Its associated density displays local self-similarity in an epsilon neighborhood of hyperbolic periodic points. The cancellation exponent of the signed measure associated with the w measure by attaching at each point the direction of the field of the asymptotic unstable eigenvectors is also analyzed. The only case for which the w-invariant measure is absolutely continuous is given by the conjugation of hyperbolic toral automorphisms with a linear automorphism. The connections with the statistical properties, and in particular with the stretching dynamics, are addressed in detail.     

Operator product expansion on a fractal: The short chain expansion for polymer networks

We prove to all orders of renormalized perturbative polymer field theory the existence of a short chain expansion applying to polymer solutions of long and short chains. For a general polymer network with long and short chains we show factorization of its partition sum by a short chain factor and a long chain factor in the short chain limit. This corresponds to an expansion for short distance along the fractal perimeter of the polymer chains connecting the network vertices and is related to a large mass expansion of field theory.

The scaling of the second virial coefficient for bimodal solutions is explained. Our method also applies to the correlations of the multifractal measure of harmonic diffusion onto an absorbing polymer. We give a result for expanding these correlations for short distance along the fractal carrier of the measure.     

SIMULATION OF MULTIPLE FRACTAL AND COMPACT GROWTH OF ULTRA-THIN FILMS ON HEXAGONAL SUBSTRATE

The multiple cluster growth of ultra-thin films with different deposition rate and different substrate temperature has been studied by kinetic Monte-Carlo simulation. With increasing diffusion rate along cluster edges (corresponding to an increasing substrate temperature), pattern structures change smoothly from fractal islands, compact islands with random shapes, to regular islands, and the average branch width of clusters increases continuously up to some constant value in the compact island limit. The formation of the multiple fractal and compact clusters can be described quantitatively by multifractal. The results of multifractal analysis show that with pattern change from fractal to compact islands, the Hausdorff dimension D₀, the information dimension D₁, and the correlation dimension D₂ decrease, while the width and height of the multifractal spectra increase.     

On a general concept of multifractality: Multifractal spectra for dimensions,entropies, and Lyapunov exponents. Multifractal rigidity

We introduce the mathematical concept of multifractality and describe various multifractal spectra for dynamical systems, including spectra for dimensions and spectra for entropies. We support the study by providing some physical motivation and describing several nontrivial examples. Among them are subshifts of finite type and one-dimensional Markov maps. An essential part of the article is devoted to the concept of multifractal rigidity. In particular, we use the multifractal spectra to obtain a "physical" classification of dynamical systems. For a class of Markov maps, we show that, if the multifractal spectra for dimensions of two maps coincide, then the maps are differentiably equivalent. (c) 1997 American Institute of Physics.