
Nicolas Pouyanne, ProfessorAssistant,
University of Versailles

"Analysis of Urn Models"

Pólya urn random processes are commonly used in theoretical computer
science, statistical physics and many other fields.
In this model, one considers an urn that contains balls of a fixed
finite number of colors.
One starts with an initial configuration.
At any step of time, one chooses one ball at random in the urn, checks
its color and puts it back into the urn.
Depending on its color, one adds new balls of different colors according
to some fixed replacement rule.
In this course, the model will be properly defined and first direct
enumerative methods will be shortly evoked.
The main topic consists in more recent and powerful approaches using
analytic combinatorics or probability theory:
generating function of successive compositions of the urn, analysis of
singularities, embedding in continuous time, martingale methods, fixed
point equations in suitable spaces of probability measures, Fourier analysis.