2024 CRM-PIMS

Summer School in Probability

There will be two main courses lasting for the entire school, given by Elliot Paquette and Perla Sousi. There will be three mini-courses, given by Emma Bailey (week 2) Jacopo Borga (week 3) Igor Kortchemski (week 4).

We expect that lectures will be streamed live. Lectures will be also be recorded and posted online.

Course descriptions

• Main course: Elliot Paquette: Random matrix theory of high-dimensional optimization Abstract
  X Optimization theory seeks to show the performance of algorithms to find the (or a) minimizer x∈ℝ^d of an objective function. The dimension of the parameter space d has long been known to be a source of difficulty in designing good algorithms and in analyzing the objective function landscape. With the rise of machine learning in recent years, this has been proven that this is a manageable problem, but why? One explanation is that this high dimensionality is simultaneously mollified by three essential types of randomness: the data are random, the optimization algorithms are stochastic gradient methods, and the model parameters are randomly initialized (and much of this randomness remains). The resulting loss surfaces defy low-dimensional intuitions, especially in nonconvex settings.

  Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

  • how random matrices can be used to describe high-dimensional inference
  • nonconvex landscape properties
  • high-dimensional limits of stochastic gradient methods.
• Main course: Perla Sousi: Random walks and branching random walks: old and new perspectives Abstract
  X This course will focus on two well-studied models of modern probability: simple symmetric and branching random walks in ℤ^d. The focus will be on the study of their trace in the regime that this is a small subset of the ambient space.

  We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

  A branching random walk (or tree indexed random walk) in ℤ^d is a non-Markovian process whose time index is a random tree. The random tree is either a critical Galton Watson tree or a critical Galton Watson tree conditioned to survive. Each edge of the tree is assigned an independent simple random walk in ℤ^d increment and the location of every vertex is given by summing all the increments along the geodesic from the root to that vertex. When $d\geq 5$, the branching random walk is transient and we will mainly focus on this regime. We will introduce the notion of branching capacity and show how it appears naturally as a suitably rescaled limit of hitting probabilities of sets. We will then use it to study covering problems analogously to the random walk case.

• Week 2: Emma Bailey: Probabilistic techniques in number theory Abstract
  X In this mini-course I will cover some classical theorems from probabilistic number theory, and then discuss some recent developments in the distribution of values of L-functions (focussing on the simplest L-function: the Riemann zeta function). I will emphasise surprising connections to random matrix theory. Many of these ideas can be visualised numerically, giving the course a computational flavour as well.
• Week 3: Jacopo Borga: Permutations in random geometry Abstract
  X I will introduce a new universal family of random permutons, called the skew Brownian permutons, describing the scaling limit of various natural models of random constrained permutations. After that, the main goal will be to discuss some connections between random permutations and random geometry. In particular, we will focus on the problem of the longest increasing subsequence in permutations sampled from the skew Brownian permuton and its connection with the study of certain directed metrics on planar maps, which conjecturally should converge in the limit to a notion of "directed Liouville quantum gravity metric."
• Week 4: Igor Kortchemski: Condensation phenomena in random trees Abstract
  X Consider a population that undergoes asexual and homogeneous reproduction over time, originating from a single individual and eventually ceasing to exist after producing a total of n individuals. What is the order of magnitude of the maximum number of children of an individual in this population when n tends to infinity? This question is equivalent to studying the largest degree of a large Bienaymé-Galton-Watson random tree. We identify a regime where a condensation phenomenon occurs, in which the second greatest degree is negligible compared to the greatest degree. The use of the "one-big jump principle" of certain random walks is a key tool for studying this phenomenon. Finally, we discuss applications of these results to other combinatorial models.