Scientific

We can classify cover times for sequences of random walks on random graphs into two types: One type is the class of cover times approximated by the maximal hitting times scaled by the logarithm of the size of vertex sets. The other type is the class of cover times approximated by the maximal hitting times. These types are characterized by the volume, effective resistances, and geometric properties of random graphs. We classify some examples, such as the supercritical Galton-Watson family trees and the incipient infinite cluster for the critical Galton-Watson family tree.

We present a modification of the IDLA model on a rotated square lattice. The process results in a forest of trees covering the upper half plane. We show an equivalence between this model and first passage percolation. We prove that with probability 1 the trees resulting from the IDLA forest are finite.

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.