2025 PIMS-CRM

Summer School in Probability

There will be two main courses lasting for the entire school, given by Tom Hutchcroft and Mathav Murugan. There will be three mini-courses, given by Nina Holden (week 1), Tianyi Zheng (week 3), and Nathanael Berestycki (week 4).

Lectures will (probably) be streamed live. Lectures recordings will also be posted on MathTube

Course descriptions

• Main course: Tom Hutchcroft: Dimension dependence of critical phenomena in percolation Abstract
  X In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter p and study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

  In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

  No prior knowledge of percolation will be assumed.

• Main course: Mathav Murugan: Heat kernel estimates and Harnack inequalities. Abstract
  X The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

  The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

  The setting for this course is a symmetric Markov process which is equivalently described using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

• Week 1: Nina Holden: Scaling limits of random planar maps Abstract
  X Planar maps are graphs embedded in the sphere such that no two edges cross, where we view two planar maps as equivalent if we can get one from the other via a continuous deformation of the sphere. Planar maps are studied in several different branches of mathematics and physics. In particular, in probability theory and theoretical physics random planar maps are used as natural models for discrete random surfaces. In this mini-course we will present scaling limit results for random planar maps and we will focus in particular on a notion of convergence known as convergence under conformal embedding. The limiting surface is a highly fractal surface called a Liouville quantum gravity (LQG) surfaces, which has its origin in string theory and conformal field theory.
• Week 3: Tianyi Zheng: Random walks on polynomial growth groups Abstract
  X Nilpotent groups are the closest class of noncommutative groups to abelian groups. Many results on Euclidean spaces can be considered there. The celebrated Gromov polynomial growth theorem asserts that a finitely generated discrete group has polynomial growth if and only if it is virtually nilpotent. More generally, for compactly generated locally compact groups of polynomial growth, structure theorems are given in a series of papers by Losert. In this minicourse, we will explore random walk models on groups of polynomial growth, starting from simple random walks on discrete groups, to more general random walks on locally compact ones, walks of unbounded range, etc. We will explain techniques to prove various estimates, limit theorems, and some applications beyond polynomial growth.
• Week 4: Nathanael Berestycki: Spectral Geometry of Liouville Quantum Gravity Abstract
  X I will discuss Liouville Brownian motion, the canonical diffusion in the random geometry defined by Liouville quantum gravity (LQG). In particular I will present some recent results on the spectral geometry of LQG, showing that the eigenvalues satisfy a Weyl law. We will also discuss a number of striking conjectures which aim to relate LQG to a phenomenon known as "quantum chaos", which will also be explained.