Mathematics

We will present a random model of population, where individuals live on several islands, and will move from one to another when they run out of resources. Our main goal is to study how the population spreads on the different islands, when the number of initial individuals and available resources tend to infinity. Finding this limit relies on asymptotics for critical random walks and (not so classical) functionals of the Brownian excursion.

The process of distribution functions of a one-dimensional super-L\'{e}vy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time-space white noises and Poisson random measures. This generalizes a recent result of Xiong (2012), where the result for a super-Brownian motion with binary branching mechanism was obtained. To establish the main result, we prove a generalized It\^o's formula for backward stochastic integrals and study the pathwise uniqueness for a general backward doubly stochastic equation with jumps. Furthermore, we also present some results on the SPDE driven by a one-sided stable noise without negative jumps. This is a joint work with Hui He and Zenghu Li.

The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.

We are interested in a particular type of subcritical Galton-Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in large conditioned non-generic trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. We investigate this phenomenon by studying scaling limits of such trees. In particular, we show that the height of such trees grows logarithmically in their size.

The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.