Past

It became apparent during the last decade that in several questions in classical complex analysis extremal configurations are fractal, making them very difficult to attack: it is not even clear how to construct or describe extremal objects. We will argue that the most promising approach is to consider conformally self-similar random configurations, which should be extremal almost surely.

Fragmentation processes model the evolution of a particle that split as time goes on. When small particles split fast enough, the fragmentation is intensive and the initial mass is reduced to dust in finite time. We encode such fragmentation into a continuum random tree (CRT) in the sense of Aldous. When the splitting times are dense near 0, the fragmentation CRT is in turn encoded into a continuous (height) function. Under some mild hypotheses, we calculate the Hausdorff dimension of the CRT, as well as the maximal H

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Given a countable set of sites S an a transition matrix p(x,y) on that set, we consider a process of particles evolving on S according to the following rule: each particle waits an exponential time and then jumps following a Markov chain governed by p(x,y); the particle keeps jumping until it reaches an empty site where it remains for another exponential time. Unlike most interacting particle systems, this process fails to

have the Feller property. This causes several technical difficulties to study it. We present a method to prove that certain measures are invariant for the process and exploit the Kolmogorov zero or one law to study some of its unusual path properties.

According to the Stokes-Einstein law, microscopic particles subject to intense bombardment by (much smaller) gas molecules perform Brownian motion with a diffusivity inversely proportion to their radius. Smoluchowski, shortly after Einstein's account of Brownian motion, used this model to explain the behaviour of a cloud of such particles when, in addition their diffusive motion, they coagulate on collision. He wrote down a system of evolution equations for the densities of particles of each size, in particular identifying the collision rate as a function of particle size.

We give a rigorous derivation of (a spatially inhomogeneous generalization of) Smoluchowski's equations, as the limit of a sequence of Brownian particle systems with coagulation on collision. The equations are shown to have a unique, mass-preserving solution. A detailed limiting picture emerges describing the ancestral spatial tree of particles making up each particle in the current population. The limit is established at the level of these trees.