50th Probability Summer School
Saint-Flour (France), 11 - 23 July 2022

Lectures 2022

Ramon Van Handel
(Princeton University, USA)
Nonasymptotic random matrix theory.

Suppose we are given a random matrix with an essentially arbitrary pattern of entry means and variances, dependencies, and distributions. What can we say about its spectrum? It may appear hopeless that anything useful can be proved at this level of generality, which lies far outside the scope of classical random matrix theory. Nonetheless, in recent years, a powerful collection of methods has emerged that provides detailed and often sharp nonasymptotic information on the spectra of essentially arbitrarily structured random matrices. The aim of my lectures is to provide an introduction to this emerging theory. The requisite background from operator theory, noncommutative probability, and high-dimensional probability will be developed along the way.

Sylvie Méléard
(École Polytechnique, Paris, France)
Interplay between scales for eco-evolutionary mathematical models.

Probabilistic models focusing on individual behaviours have been developed over the past fifteen years, to model eco-evolution problems and understand the subtle scale relationships that may exist between genetic and demographic parameters and that impact the model's behaviour and its macroscopic approximations.

In this course, we will develop this point of view around 3 main themes: modeling of phenotypically structured populations and eco-evolutionary phenomena; identification of the underlying phylogenies in a population with a trait structure and interactions or impacted by a gradual change in environment; population extinction and quasi-stationary distributions.In each of these areas, we will show how surprising behaviours emerge from this interplay of scales and we will provide quantitative answers. These lectures will show how biological questions can lead to rich and original mathematics.

The structured population dynamics are modeled by measure-valued processes describing the individual behaviors, including birth and death and interactions between individuals and possible mutations. The structure depends on a quantitative parameter describing the genetic or phenotypic characteristics of each individual. These processes are then infinite dimensional generalizations of classical birth and death processes. From these discrete processes we can deduce different approximations according to the scale ratio between the different parameters, mainly the population size and the mutation rates. Some specific scalings lead to original convergence theorems and new techniques of proofs. We will present such results in the first part of the lectures. In the second part of the lectures, we observe the population at a fixed time and we want to find the genetic history of the individuals alive at this time. In population genetics, this is captured by the coalescent process in duality with the branching processes. In our purpose, we add non constant populations and interactions between individuals. We will see different mathematical approaches to overcome these difficulties.
In the third part of the lectures, the population is composed of a finite number of types and modeled by a finite-dimensional birth and death process. We assume that it goes almost surely to extinction. The size of the birth and death processes is parameterized by a scaling parameter which tends to infinity. We will show how to quantify the extinction time in function of this parameter and how to relate the quasi-stationary distribution of the process to the dynamical system which approximates the process in finite time.

Allan Sly
(Princeton University, USA)
Random Constraint Satisfaction Problems.

Random constraint satisfaction problems, such as the random K-SAT model and colourings of random graphs naturally emerge in the study of combinatorics and theoretical computer science. Ideas from statistical physics predict a series of phase transitions these models undergo as the density of constraints increases. These describe both qualitatively and quantitatively the number of solutions, the geometry of the space of solutions and threshold for when solutions exist.

The lectures will begin with a study of random graphs and spin systems on trees and locally treelike graphs. Then we will investigate a series of heuristics based on belief propagation, a message passing algorithm, used by physicists to predict the behaviour of random CSPs. A major focus will be on how, by encoding fixed point messages combinatorially, many of these predictions can be rigorously verified.

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