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Jean Bertoin, Laboratoire de Probabilités,
Université Paris 6
Some aspects of self-similar fragmentations |
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Fragmentation phenomena can be observed in many sciences
at a great variety of scales. To give just a few examples,
let us simply mention the studies of stellar fragments
and meteoroids in astrophysics, fractures and
earthquakes in geophysics, breaking of crystals in
crystallography, degradation of large polymer chains
in chemistry, fission of atoms in nuclear physics,
fragmentation of a hard drive in computer science, ...,
not to mention crushing in the mining industry.
The purpose of this minicourse is to introduce a mathematical framework
which might serve as model for situations in which fragmentations
occurs randomly and repeatedly as time passes.
We shall first construct a large family of fragmentation processes
in connection with branching Markov chains in continuous times.
Then, we shall investigate fine properties of such processes depending on
their characteristics. In particular, we shall point at the phenomenon
of formation of dust when the index of self-similarity is negative,
and to a homogeneization type property for positive indices.
Files:
Course
Exercises
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Olle Haggstrom
Percolation theory: the number of infinite clusters |
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One of the most important and celebrated theorems in percolation theory
is the uniqueness of the infinite cluster for i.i.d. percolation on Z^d.
Although this result dates back to the late 1980's, the issue of
uniqueness of infinite clusters is still very much alive as a research
area. For instance, what happens when we move to various types of
dependent models, or to other lattices? And what happens if we focus
not on connected components, but instead on entangled components, or
on rigid components? Some answers to these questions will be given.
Files:
Johan Jonasson and Olle Haggstrom -
Uniqueness and non-uniqueness in percolation theory
Olle Haggstrom (2003) Uniqueness of
infinite rigid components in percolation models: the case of nonplanar
lattices PTRF 127 513-534
Exercises
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| Webmaster:
Ivan Neto |
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8th School
