On a symmetry of turbulence

This paper presents results on symmetries of the spectrum of singularities for random cascades found in the statistical theory of turbulence. It is shown that empirical dimension curves possess a natural symmetry whose presence restricts the class of allowable probability distributions of the cascade generator in a simple manner. In particular, necessary and sufficient conditions on the probability distribution of the generator are obtained for symmetry of the singularity spectrum within a large class of cascade models.     

On weighted heights of random trees

Consider the family treeT of a branching process starting from a single progenitor and conditioned to havev=v(T) edges (total progeny). To each edge <e> we associate a weightW(e). The weights are i.i.d. random variables and independent ofT. The weighted height of a self-avoiding path inT starting at the root is the sum of the weights associated with the path. We are interested in the asymptotic distribution of the maximum weighted path height in the limit asv=n. Depending on the tail of the weight distribution, we obtain the limit in three cases. In particular ify ² P(W(e)> y)0, then the limit distribution depends strongly on the tree and, in fact, is the distribution of the maximum of a Brownian excursion. If the tail of the weight distribution is regularly varying with exponent 0<2, then the weight swamps the tree and the answer is the asymptotic distribution of the maximum edge weight in the tree. There is a borderline case, namely,P(W(e)> y)cy ^–2 asy, in which the limit distribution exists but involves both the tree and the weights in a more complicated way.     

Independent random cascades on Galton-Watson trees

Consider an independent random cascade acting on the positive Borel measures defined on the boundary of a Galton-Watson tree. Assuming an offspring distribution with finite moments of all orders, J. Peyrière computed the fine scale structure of an independent random cascade on Galton-Watson trees. In this paper we use developments in the cascade theory to relax and clarify the moment assumptions on the offspring distribution. Moreover a larger class of initial measures is covered and, as a result, it is shown that it is the Hölder exponent of the initial measure which is the critical parameter in the Peyrière theory.

     

Scaling limits for point random fields

If X is a point random field on $\text{R}$^d then convergence in distribution of the renormalization C_λ|X_λ ? α_λ| as λ → ∞ to generalized random fields is examined, where C_λ > 0, α_λ are real numbers for λ > 0, and X_λ(f) = λ^?dX(f_λ) for $\text{f}{}_{\mathrm{\lambda }}\text{(x) = f(}\text{x}\text{λ}\text{)}$. If such a scaling limit exists then C_λ = λ^θg(λ), where g is a slowly varying function, and the scaling limit is self-similar with exponent θ. The classical case occurs when $\text{θ =}\text{d}\text{2}$ and the limit process is a Gaussian white noise. Scaling limits of subordinated Poisson (doubly stochastic) point random fields are calculated in terms of the scaling limit of the environment (driving random field). If the exponent of the scaling limit is $\text{θ =}\text{d}\text{2}$ then the limit is an independent sum of the scaling limit of the environment and a Gaussian white noise. If $\text{θ <}\text{d}\text{2}$ the scaling limit coincides with that of the environment while if $\text{θ >}\text{d}\text{2}$ the limit is Gaussian white noise. Analogous results are derived for cluster processes as well.     

Infinite divisibility of a bethe lattice ising model

It is shown that the probability distribution for the infinite-volume, free-boundary-condition Ising ferromagnet on the Bethe lattice under zero external field is infinitely divisible with respect to the group operation of pointwise multiplication of spin variables.     

A cascade decomposition theory with applications to Markov and exchangeable cascades

A multiplicative random cascade refers to a positive -martingale in the sense of Kahane on the ultrametric space A new approach to the study of multiplicative cascades is introduced. The methods apply broadly to the problems of: (i) non-degeneracy criterion, (ii) dimension spectra of carrying sets, and (iii) divergence of moments criterion. Specific applications are given to cascades generated by Markov and exchangeable processes, as well as to homogeneous independent cascades.