Kaisa_2012_3_photo by Veikko Somerpuro

Phase transintions and critical phenomena

We will study several probabilistic models of systems comprised of a large number of small interacting parts. A focus of the study will be on phase transition, that is, abrupt change of the behaviour of the system as under small change of parameters. We will study, in particular, the following systems:

- in the model of percolation, one studies connectivity properties of random networks obtained by declaring edges of a lattice open or closed with some probability, independent of each other. This classical setup models behaviour of porous media, emergence of forest fires and epidemics.

- the Ising model is a classical model of statistical physics, explaining the transition for ferromagnetic to paramagnetic behaviour in metals at high temperature.

- the model of self-avoiding walks aims at explaining mathematically the transition of polymers (e. g., proteins) from the folded phase to unfolded one, i. e., extended in space.

We will prove basic general properties of these and other models, in particular, the existence of phase transition. We will also look more closely at when and how the transition occurs. A particularly precise information, including exact results, can be given in dimension 2.

The course will be mostly self-contained; understanding elementary probability should be enough to follow. In particular, the course is independent of the "Topics in probability I" course.

The intended content of the course.

- introduction;
- combinatorics of percolation, spin models and random cluster models;
- Peierls argument: existence of phase transition;
- correlation inequalities;
- planar duality and Russo-Seymour-Welsh theory;
- Kesten's theorem: the critical temperature for bond percolation on the square lattice is 1/2;
- Burton-Kean argument for the uniqueness of the infinite cluster;
- exponential tails for cluster size in sub-critical percolation;
- star-triangle transformation; computation of critical temperature for planar lattices;
- the critical temperature for the Ising model and for self-awoiding walks on hexagonal lattice;
- conformal invariance for critical percolation; Cardy-Smirnov formula;
- conformal invariance in the Ising model.

12.2.2018 klo 12:00 - 3.5.2018 klo 23:59


Tästä osiosta löydät kurssin opetusaikataulun. Tarkista mahdolliset muut aikataulut kuvauksesta.

Ma 12.3.2018
10:15 - 12:00
To 15.3.2018
10:15 - 12:00
Ma 19.3.2018
10:15 - 12:00
To 22.3.2018
10:15 - 12:00
Ma 26.3.2018
10:15 - 12:00
To 5.4.2018
10:15 - 12:00
Ma 9.4.2018
10:15 - 12:00
To 12.4.2018
10:15 - 12:00
Ma 16.4.2018
10:15 - 12:00
To 19.4.2018
10:15 - 12:00
Ma 23.4.2018
10:15 - 12:00
To 26.4.2018
10:15 - 12:00
Ma 30.4.2018
10:15 - 12:00
To 3.5.2018
10:15 - 12:00

Muu opetus

21.03. - 28.03.2018 Ke 10.15-12.00
11.04. - 02.05.2018 Ke 10.15-12.00
Konstantin Izyurov
Opetuskieli: englanti


Exercise sheet 1


Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Probability theory I,II, and their prerequisites

A collection of topics in probability and/or stochastic processes

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Some of the following: general aspects of stochastic processes, Markov chains and processes, Brownian motion, ergodic theory, large deviations and concentration of measure, geometric probability, integrable probability, point processes and determinantal processes, random graphs random graphs


Lectures and exercise classes

Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later