Minicourse I - Tom Hutchcroft

Title: Dimension dependence of critical phenomena in percolation

Abstract: 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 an infinite cluster emerging at a critical probability pc(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 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.

Minicourse II - Antonio Auffinger

Title: Mathematical foundations of spin glass theory

Abstract: The study of spin glass models has led to a far-reaching and beautiful physical theory whose techniques have been applied to a large collection of problems in theoretical computer science, statistics, optimization and biology. Many of the physical predictions can be formulated as purely mathematical questions, often extremely hard, about large random systems in high dimensions. In recent decades, a new and exciting area of research has emerged in probability theory around these problems. While their original aim was to explain the strange behavior of certain magnetic alloys, the study of spin glass models has led to a far-reaching and beautiful physical theory whose techniques have been applied to a myriad of problems in theoretical computer science, statistics, optimization and biology. As many of the physical predictions can be formulated as purely mathematical questions, often extremely hard, about large random systems in high dimensions, in recent decades a new area of research has emerged in probability theory around these problems. 

Mathematically, a mean-field spin glass model is a Gaussian process (random function) on the discrete hypercube or the sphere in high dimensions. A fundamental challenge in their analysis is, roughly speaking, to understand the size and structure of their super-level sets as the dimension tends to infinity, which are often studied through smooth objects like the free energy and Gibbs measure whose origin is in statistical physics. The aim of the summer school is to introduce students to landmark results on the latter while emphasizing the techniques and ideas that were developed to obtain them, as well as exposing the students to some recent research topics.

Prerequisites: The knowledge that is expected from the students is the material covered in first-year courses in probability theory and analysis. More advanced material in probability is helpful but not required as we will review what we need in class. As such, the course is well suited for any graduate student in mathematics with interest in probability or mathematical physics.

References: Students are expected to be familiar with concepts covered in first-year graduate courses in analysis and probability. A good preparation includes Chapter 1 (Measure Theory), Chapter 2 (Laws of Large Numbers) and 3 (Central Limit Theory) in Probability: Theory and Examples, 5th ed, by R. Durrett (2019)