From Statistical Physics to Random Geometry

Abstract:

The goal of this course will be to give an introduction to spin systems defined on a lattice Z^d with an underlying continuous symmetry. This includes the following examples:

- The Gaussian Free Field whose 'spins' are R valued

- The Villain and XY models whose spins are S^1 valued

- The classical Heisenberg model with values in S^2

- Lattice gauge theory with continuous gauge group G

Some of the classical techniques which are very powerful when dealing with a discrete symmetry spin system (for example the so-called Peierls argument for the Ising model) do not apply for continuous symmetry spin systems. This course will explain how such continuous symmetries affect the fluctuations in the system and will introduce some of the main relevant techniques, among which:

- Mermin-Wagner theorem (on the absence of long-range order in 2d)

- Reflection positivity

- duality transformations

- topological defects

- techniques from Bayesian statistics

Abstract:

Stochastic topology is the study of random topological spaces. Topological statistical mechanics is the study of complicated configuration spaces which arise naturally in physics as energy landscapes. In both settings, we might be particularly interested in asymptotic behavior of the topology as some parameter tends to infinity. A related common theme is the idea of a phase transition, where the qualitative behavior of the system shifts drastically with small variance of a parameter. In this mini-course, we will survey recent developments in these areas, and some directions for future study will be pointed out along the way.

Slides of first lecture

Abstract:

In planar random geometry, a plethora of conformally invariant and covariant objects has been emerged in the recent years. Among these particularly fruitful have been random fractal curves derived from one-dimensional Brownian motion: Schramm-Loewner Evolutions, Conformal Loop Ensembles, and their variants. Originally they were introduced in the context of critical models in statistical physics to understand conformal invariance and critical phenomena. Such curves describe scaling limits of lattice interfaces, and level and flow lines of random fields, but have also turned out to be quite interesting by their own right. In these lectures, I will introduce these models for conformally invariant random paths and loops, discuss their relation to critical models, some of their geometric properties, and consequences.

Lecture notes