There will be two main courses lasting for the entire school, and three mini-courses, Lectures will be taped and posted on MathTube.
Ivan Corwin: Interacting particle systems, growth models, stochastic PDEs and directed polymers through the lens of the stochastic six vertex model.
Introduced by Gwa and Spohn in 1992, the stochastic six vertex model is a cousin of the symmetric six vertex model introduced much earlier by Pauling in 1935. Through limit transitions and various specialization of parameters, this models can be related to a rich hierarchy of probabilistic systems including the asymmetric simple exclusion process, Kardar-Parisi-Zhang (KPZ) stochastic PDE, stochastic telegraph equation, and certain special models of random walks in random environments, directed polymers and last passage percolation.
This course will focus on advances in our understanding of the above models, through the lens of the stochastic six vertex model. We will explain how old tools such as the Bethe ansatz, Yang-Baxter equation, fusion, symmetric functions and Markov dualities have gained new probabilistic applications and interpretations. Using these tools we will extract precise asymptotics of these systems related to both the Kardar-Parisi-Zhang universality class and stochastic PDE, and uncover other remarkable probabilistic properties.
Frank den Hollander: Metastability for interacting particle systems
Metastability is a wide-spread phenomenon in the dynamics of non-linear systems subject to the action of temporal random forces, typically referred to as noise. In the narrower perspective of statistical physics, metastable behaviour can be seen as the dynamical manifestation of a first-order phase transition, i.e., a crossover that involves a jump in some intrinsic physical parameter, such as the energy density or the magnetisation. Attempts to understand and model metastable systems mathematically go back to the 1930's. The modern mathematical approach to metastability dates from around 1970.
One approach to metastability is via the theory of large deviations in path space. The realisation that metastable behaviour is controlled by large deviations of the random processes driving the dynamics has permeated most of the mathematical literature on the subject. The present mini-course focusses on an alternative way to tackle metastability, which initiated around 2000. It interprets the metastability phenomenon as a sequence of visits of the path to different metastable sets, and focuses on the precise analysis of the respective hitting probabilities and hitting times of these sets with the help of potential theory. The key point is the realisation that, in the specific setting related to metastability, most questions of interest can be reduced to the computation of capacities, and that these capacities in turn can be estimated by exploiting variational principles. In this way, the metastable dynamics of the system can essentially be understood via an analysis of its statics. This constitutes a major simplification, and acts as a guiding principle.
The setting of this mini-course is the theory of reversible Markov processes. Within this limitation, there is a wide range of models that are adequate to describe a variety of different systems. The models we aim at range from finite-state Markov chains, finite-dimensional diffusions and stochastic partial differential equations, via mean-field dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and probabilistic cellular automata. Our aim is to unveil the common universal features of these systems with respect to their metastable behaviour. Along the way we will encounter a variety of ideas and techniques from probability theory, analysis and combinatorics, including martingale theory, variational calculus and isoperimetric inequalities. It is the combination of physical insight and mathematical tools that allows for making progress, in the best of the tradition of mathematical physics.
- Paul Bourgade: Branching processes in random matrix theory and number theory (week 1)
I will first review the principles to characterize extrema for branching processes. I will then focus on implementation of these methods for predictions by Fyodorov, Hiary and Keating, which link the maximum of random characteristic polynomials and the maximum of the Riemann zeta function on most short intervals along the critical line.
- Jean-Francois Le Gall: Brownian geometry (week 4)
We discuss the discrete and continuous models of random geometry that have been studied extensively in the recent years. Planar maps, which are finite connected graphs embedded in the sphere, are the basic discrete models and are usually chosen uniformly at random in a given class, for instance the class of all triangulations with a fixed number of faces. The so-called Brownian sphere, or Brownian map, arises as the universal scaling limit of random planar maps equipped with the usual graph distance, in the Gromov-Hausdorff topology. The case of planar maps with a boundary leads to Brownian disks, which also appear as special subsets of the Brownian sphere. The connections between Brownian disks and the Brownien sphere are best understood via a new construction of Brownian disks from excursion theory for Brownian motion indexed by the Brownian tree.
- Nike Sun: Phase transitions in random constraint satisfaction problems (week 2)
We will describe recent progress in determination of asymptotic behavior in random constraint satisfaction problems, including the independent set problem on random graphs, random regular NAE-SAT, and random SAT. The results include sharp phase transitions and some understanding of solution geometry, particularly in the setting of the random regular NAE-SAT problem. In this lecture we will survey the physics heuristics, and explain how they lead to combinatorial models for the solution geometry, which form a basis of mathematical approaches to these problems.
We will then go over some of the mathematical approaches to the study of random CSPs, particularly involving the (first and second) moment method. We will see that implementing the moment method often reduces to the solution of difficult (non-convex) optimization problems. We will discuss one basic strategy that has been used to solve some of these problems, which is to (1) use "a priori" estimates to localize the optimization to a small neighborhood, and (2) use the contractivity of tree recursions and a "local update" procedure to solve the optimization within the small neighborhood.
These lectures are based in part on joint works with Zsolt Bartha, Jian Ding, Allan Sly, and Yumeng Zhang.