2027 CRM-Fields-PIMS

Summer School in Probability

There will be two main courses lasting for the entire school, given by Roland Bauerschmidt and Nina Gantert. There will be three mini-courses, given by TBA.

Lectures will be recorded and streamed online.

Course descriptions

• Main course: Roland Bauerschmidt: Probabilitistic methods in quantum field theory Abstract Notes.
  X The first week will cover discrete models of quantum and statistical field theory such as the Ising and the O(n) models on the hypercubic lattice. Specific topics covered include the Markov property and reflection positivity, the transfer matrix, and their consequences including phase transitions. The relation to quantum mechanics will be discussed.

  The second week will focus on the Gaussian free field, both on the lattice and in the continuum. Specific topics covered include the covariance structure of the free field, Wick's theorem, and the Fock space. The focus will be on the probabilistic point of view, but the quantum interpretation will be discussed. More advanced aspects such as hypercontractivity and regularity of the free field in various function spaces as well as probabilistic decompositions of the free field will be covered.

  The third and fourth weeks focus on quantum field theory in the continuum. After the introduction of the axioms of Euclidean quantum field theory, several examples such as the sine- and sinh-Gordon model and the Phi^4 model will be studied from different perspectives -- probabilistic and analytic.

• Main course: Nina Gantert: Branching random walks Abstract Notes
  X Branching random walk and its continuous counterpart, branching Brownian motion, are fundamental objects of probability theory. They are interesting for several reasons:

  • they are a key example of random recursive structures
  • they are models for population growth and spatial displacements
  • there are well-known connections to partial differential equations, in particular the KPP equation
  • the study of extrema of branching random walks has much in common with the investigation of extrema for other models as the Gaussian free field

  In the first lectures we will introduce branching random walks and branching Markov chains and discuss recurrence, transience and the linear growth of the maximal distance to the origin. Important tools that we will explain are the use of large deviation theory and the first and second moment method and ballot theorems, used to study the extrema of branching Brownian motion. We will also introduce branching Brownian motion and explain its connection to the KPP equation.

  We will then move to more recent directions: branching random walks with annihilation, branching random walks in random environments and branching random walks with catalysts. Here, some tools from the theory of random environments come into play, and we will discuss regeneration times and renormalization techniques, as well as some methods from interacting particle systems.

  There will be an emphasis on open problems which we will mention throughout the lectures. Prerequisite is knowledge of general Probability Theory (conditional expectation, martingales, Brownian motion).

• Week 1: name: title Abstract
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• Week 2: name: title Abstract
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• Week 4: name: title Abstract Notes
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