# Master Class in Planar Statistical Physics 2015/16

### 14 Sep 2015

**Video lectures: UNIGE mediaserver**

**Description of the programme:**

Amazing progress has been made in the understanding of planar statistical physics during recent years. The introduction of the Schramm-Loewner Evolution and the developments in the theory of random planar maps and lattice models enabled mathematicians to connect the rigorous approach to statistical physics with the traditional approach from physics based on exact integrability and conformal field theory. The goal of this programme is to provide courses on recent advancements in this field. More general courses on probability will also be provided during the first term.

The program is aimed at Master students, though advanced undergraduates and beginning PhD's are also welcome to apply. The participants will enroll in a one-year master program at the University of Geneva starting in September 2015, providing 60 ECTS credits. If they wish, participants will be offered the possibility of finishing a Masters degree from the University of Geneva by completing a Master thesis for 30 additional ECTS credits. A number of fellowships covering accommodation and local expenses are available.

**Registration:**

For full consideration, candidates should apply before the 17th of January 2015 by filling out the following form:

*REGISTRATION FORM - Submissions are closed*

Late applications will be considered subject to availability of resourses.

Questions can be addressed to the following email address.

**List of confirmed lecturers:**

I. Benjamini

D. Chelkak

D. Cimasoni

N. Curien

H. Duminil-Copin

C. Hongler

J. Miller

S. Smirnov

Y. Velenik

W. Werner

**Preliminary list of courses **

**First semester courses:**

1. Introduction to statistical mechanics (Velenik)

The aim of this course is to introduce the students to the mathematical analysis of lattice spin systems. Several fundamental questions will be addressed and illustrated in the simplest relevant situations. Among others, the following topics will be covered:

- The phase diagram of the Ising model: correlation inequalities, infinite-volume Gibbs states, uniqueness and non-uniqueness, etc.
- The discrete Gaussian Free Field: random walk representation, existence/nonexistence of infinite-volume Gibbs states, etc.
- Two-dimensional models with continuous symmetry: the Mermin-Wagner theorem, decay of correlations, etc.

The course will be based on an ongoing book, an early version of which can be found at http://www.unige.ch/math/folks/velenik/smbook/index.html.

2. Brownian Motion and stochastic calculus

We will discuss some basic objects of stochastic analysis. The following topics will for instance be covered: Brownian motion (construction and path properties), stochastic integration, Ito's formula and applications, stochastic differential equations and their links to partial differential equations.

3. Martingales and Markov processes

This course will be devoted to the study of martingales and Markov processes, both in discrete and continuous time, with an emphasis on the interplay of the two objects. We will start with the theory of martingales, including inequalities, stopping time and convergence theorems. We will then study Markov processes, including state classification, invariant measures, and infinitesimal generators.

4. On various aspects of the dimer and planar Ising models (D. Cimasoni)

The aim of this course is to study the dimer (and to a lesser extend, Ising) models on 2-dimensional lattices, i.e. graphs embedded in surfaces. The focus will be put on the amazing variety of tools coming into play in this study, tools from areas of mathematics as diverse as combinatorics, algebraic geometry, and algebraic topology. We shall introduce these tools along the way when we need them, so no previous knowledge of these theories will be assumed.

More precisely, we plan to cover the following topics:

- Combinatorics of dimer configurations on abstract graphs.
- The Kasteleyn theorem for dimers on planar graphs, and its extension to surface graphs.
- Dimer models on bipartite biperiodic graphs.
- Mappings between the dimer and Ising models.
- The Kac-Ward formula for the Ising model on planar graphs, and its extension to surface graphs.

If time permits, we will use all of this to take a closer look at these models at criticality.

**Second semester courses:**

5. Conformal invariance of lattice models (S. Smirnov)

This class will be devoted to the proof of conformal invariance of several models of planar statistical physics, including the dimer, percolation and Ising models.

6. Schramm-Loewner Evolution and Gaussian Free Field (W. Werner)

Building on the courses of the first semester and on some basic complex analysis that we shall first recall, we will give an introduction to the Schramm-Loewner Evolutions, and exhibit some of the relations between these remarkable random planar curves with the Gaussian Free Field, with the scaling limits of interfaces for critical percolation and for the Ising model, and with the uniform spanning trees.

7. Random planar maps

The goal of this class is to present the theory of large random planar maps. Among others, we will treat the question of the enumeration of planar maps, of their local and scaling limits, and recent progress concerning their connection with Liouville Quantum Gravity.

8. Geometric representations of lattice models (H. Duminil-Copin)

This class is a continuation of the class "Introduction to statistical mechanics". Geometric representations have been used to study spin-spin correlations of spin models on lattices. These models include:

- Bernoulli percolation
- Fortuin-Kasteleyn percolations (also called random-cluster models)
- the loop O(n) models
- the high and low temperature of the Ising model
- the random-current representations of the Ising model.

This course will be devoted to the study of these classical geometric representations and to the applications of these models to spin models.

**A number of mini-courses including:**

9. Coarse geometry and random processes (I. Benjamini)

We will see how geometric properties of infinite graphs, such as isoperimetry, hyperbolicity and symmetry ( as well stationarity, which is random version of symmetry) are manifested in the behavior of random processes, such as random walks, harmonic functions and percolation, on the graphs.

10. Probability on groups

This class is a continuation of the class "coarse geometry and random processes". We will discuss the connection between the properties of a group and of statistical models defined on Cayley graphs of the group.

11. Large deviations

This class covers the general large deviation theory, the relevant convex analysis, and the large deviations of i.i.d. processes on three levels: Cramer's theorem, Sanov's theorem, and the large deviation principle for trajectories of random processes.