in Probability 2021

There will be two main courses lasting for the entire school, given in person at CRM, and three mini-courses given remotely. Non-local participants can participate in all lectures virtually. Lectures will be taped and posted online.

Course website and Course notes (in progress), lecture videos.

Combinatorial optimization is concerned with optimization problems: one may be attempting to minimize the total distance travelled when visiting a fixed set of cities (the travelling salesman problem) or the total miles of asphalt required to ensure that one can drive from any city to any other (the minimum spanning tree problem). Remarkably, the probabilistic analysis of some such models leads directly to the heart of modern discrete probability, in many cases to the very same questions that arise from attempting to describe and analyze physical, biological and social systems.

This course will provide an introduction to the probabilistic analysis of optimization problems in large discrete systems, including: random minimum spanning trees; first-passage percolation on trees and graphs; and mixing times of interacting particle systems. It will also cover analytic tools for describing asymptotic behaviour and limit objects (hydrodynamic limits, graph limits) and proving universality (contraction methods, Stein's method).

Course website and course materials, lecture videos.

The goal of statistical mechanics is to describe the large-scale behavior of collections of simple elements, often called spins, that interact through locally simple rules and are influenced by some amount of noise. We will focus on the situation where the local interactions are chosen at random, in which case the models are usually called "spin glasses". Such models are already surprisingly difficult to analyze when all spins interact with each other. In this course, we will revisit this analysis using tools from the theory of Hamilton-Jacobi equations.

After an overview of the material covered in the course, we will start by focusing on the very simple Curie-Weiss model. Along the way, we will introduce analytical techniques related to the study of Hamilton-Jacobi equations, and use them to identify the limit free energy of the model. We will next transpose this strategy into a first class of disordered models coming from statistical inference. In terms of difficulty, these models are a useful bridge between the Curie-Weiss model and spin glasses. We will finally turn our attention to the latter models, in which infinite-dimensional Hamilton-Jacobi equations arise.

Lecture notes and Glossary, lecture videos.

Branching-selection systems are a class of particle systems introduced by physicists as toy models for populations under natural selection. In such a particle system, each particle has a 'fitness' value determined by its spatial location. Whenever a particle branches (produces an offspring particle), the particle in the system with lowest fitness value is killed, keeping the total number of particles constant. It turns out that over short timescales, these particle systems can be approximated by partial differential equations known as free boundary problems. We will see how these results can be proved, and how (at least in some cases) they can be used to determine the long-term behaviour of the particle system. We will also discuss open conjectures about universal behaviour of the genealogy (family tree) of a sample of particles.

Resources:
References,
Slides from Lecture 1,
Lecture 2,
Lecture 3,

Handout on local weak limits,
Exercises:
Set 1,
Set 2,
lecture videos.

Large systems of stochastically interacting particles in which the infinitesimal evolution of each particle depends on the states of neighboring particles with respect to an underlying interaction graph are fundamental stochastic processes that model phenomena in a wide range of disciplines, including statistical mechanics, epidemiology and math finance. When the graph is complete, it is well known that the dynamics of a typical particle converges, as the size of the graph goes to infinity, to a stochastic process called the mean-field limit. In this mini-course, we will focus on the complementary case when the underlying graph is sparse (that is, having uniformly bounded (average) degree) and describe what is known about the asymptotics of the empirical measure and the limiting dynamics of a typical particle for suitable sequences of (random) graphs. Throughout, we will provide illustrative examples, and mention open questions.