Related papers: An introduction to Mandelbrot cascades

Linear rate equations are used to describe the cascading decay of an initial heavy cluster into fragments. Using a procedure inspired by the similar, but continuous case of jet fragmentation in QCD, this discretized process may be analyzed…

Our goal is to obtain a complete set of angular observables arising in a generic multi-body process. We show how this can be achieved without the need to carry out a likelihood fit of the angular distribution to the measured events.…

This paper shows that an arbitrary generic submanifold in a complex manifold can be deformed into a 1-parameter family of generic submanifolds satisfying strong nondegeneracy conditions. The proofs use a careful analysis of the jet spaces…

The motivation of this article is to estimate multifractality classification and model selection parameters: the first-order scaling exponent $c_1$ and the second-order scaling exponent (or intermittency coefficient) $c_2$. These exponents…

We consider fractal percolation (or Mandelbrot percolation) which is one of the most well studied example of random Cantor sets. Rams and the first author studied the projections (orthogonal, radial and co-radial) of fractal percolation…

We consider the problem of locating the source of a network cascade, given a noisy time-series of network data. Initially, the cascade starts with one unknown, affected vertex and spreads deterministically at each time step. The goal is to…

We study small deviations in Mandelbrot cascades and some related models. Denoting by $Y$ the total mass variable of a Mandelbrot cascade generated by $W$, we show that if $\log \log 1/P(W \leq x) \sim \gamma \log \log 1/x$ as $x \to 0$…

We first give a general introduction to the mean curvature flow, and then discuss fundamental results established over the last 10 years that yield a precise theory for the flow through singularities in $\mathbb{R}^3$. With the aim of…

This announcement describes a probabilistic approach to cascades which, in addition to providing an entirely probabilistic proof of the Kahane-Peyri\`ere theorem for independent cascades, readily applies to general dependent cascades.…

Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To…

In this Note, we present recent developments in the Renormalization Theory of quadratic polynomials and discuss their applications, with an emphasis on the MLC conjecture, the problem of local connectivity of the Mandelbrot set, and on its…

We obtain four extended Newton's methods and three extended Mandelbrot's recurrence formulas from the Wasan (Japanese mathematics in the Edo period (1603-1868)). Furthermore, two extended Newton's methods relate to one of the extended…

A feasible model is introduced that manifests phenomena intrinsic to iterative complex analytic maps (such as the Mandelbrot set and Julia sets). The system is composed of two coupled alternately excited oscillators (or self-sustained…

The problem of intermittency in developed hydrodynamic turbulence is considered. Explicit formulae taking into account effects of finite size of the inertial range are presented for the whole set of intermittency exponents. The formulae fit…

We prove existence of all moments of the multiplicative coalescent at all times. We obtain as byproducts a number of related results which could be of general interest. In particular, we show the finiteness of the second moment of the $l^2$…

We prove the decoration theorem for the Mandelbrot set (and Multibrot sets) which says that when a "little Mandelbrot set" is removed from the Mandelbrot set, then most of the resulting connected components have small diameters.

We provide necessary and sufficient conditions for explosion and implosion of birth-and-death (non-Markov) continuous-time random walks. In other words, we obtain conditions for $\infty$ to be accessible and for it to be an entrance point.…

We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk…

This note is mainly to point out, if needed, that uncertainty about models and their parameters has little to do with a `paradox'. The proposed `solution' is to formulate practical questions instead of seeking refuge into abstract…

We investigate the Fourier dimension, $\dim_F\mu$, of Mandelbrot multiplicative cascade measures $\mu$ on the $d$-dimensional unit cube. We show that if $\mu$ is the cascade measure generated by a sub-exponential random variable then…