Cursos

Los cursillistas invitados son:

Dr. Nicolas Curien. Université Paris-Sud Orsay, Francia

Scaling limits for branching processes with integer types

Discrete growth-fragmentation trees are plane trees where each vertex carries a non-negative integer label. They generalize the simply generated trees and have appeared recently in many places in discrete random geometry: random split trees, conditioned Bienaymé--Galton--Watson trees, peeling trees in random planar maps, random fully parked trees... Under the assumption that the rescaled types along a branch converge towards a positive self-similar Markov process, those random discrete labeled trees converge in the scaling limit towards Bertoin's growth-fragmentation trees. The course will be devoted to set the basics of this theory and to derive a few geometric consequences of the scaling limit results.

Notas del curso!!!

Dra. Laura Eslava. IIMAS, UNAM, México

Critical percolation and emergence of a giant component

An interesting phenomenon of increasing graph processes is the existence of a time window that marks the emergence of a connected component whose size is comparable with the size of the underlying graph of the process. This phenomenon is universal across several models; that is, models with distinct underlying graphs or edge-addition rules present the same qualitative characteristics. Instead, the differences between such models are reflected in the quantitative description of both the location and width of the time window, as well as the growth rate of the giant component right after its emergence.

In this course, we will review some of the techniques used in this area and heuristics that explain (in some cases) how the local structure of the underlying graph determines the quantitative characteristics of this phenomenon. We will mainly work with two models: percolation on the hypercube and a graph process where the edge-addition rule generates non-trivial dependencies throughout the evolution of the process.

Notas del curso!!!

Dr. Avelio Sepúlveda, Universidad de Chile, Chile

The Gaussian free field and its two-valued sets

In this course, we will study the two-dimensional continuum Gaussian free field (GFF). The GFF is the generalisation of Brownian motion when the time set is now a two-dimensional open set $D$. The main technical difficulty of this field is that it is no longer an L^2 function and it has to be defined as a generalised function, i.e., (Schwartz) distribution. The main objective of this course is to describe the Markov property of this field and show the existence of the analogue of first exit times: the two-valued sets.