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250093 VO Lattice Models (2022S)

3.00 ECTS (2.00 SWS), SPL 25 - Mathematik

Moodle

Registration/Deregistration

Note: The time of your registration within the registration period has no effect on the allocation of places (no first come, first served).

Details

Language: English

Examination dates

Lecturers

Alexander Glazman

Classes (iCal) - next class is marked with N

The course will be on site (hopefully), but I’m planning to do it on my IPad+projector and could link it to zoom. So it will be possible to follow the lectures online and there will be a recording + lecture notes.

Wednesday 02.03. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 09.03. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 16.03. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 23.03. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 30.03. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 06.04. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 27.04. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 04.05. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 11.05. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 18.05. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 25.05. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 01.06. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 08.06. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 15.06. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 22.06. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 29.06. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock

Information

Aims, contents and method of the course

Phase transitions are natural phenomena in which a small change in an external parameter, like temperature or pressure, causes a dramatic change in the qualitative structure of the object. To study this, many scientists (such as Nobel laureates Pauling and Flory) proposed the abstract framework of lattice models: the material was modeled as a collection of particles on a regular lattice, interacting only with their nearest neighbors. In spite of the simplistic nature of this assumption, lattice models have proven to be a rich laboratory for the mathematical study of phase transitions. Since the revolutionary work of Schramm in 2000, the probabilistic approach to the study of these models has yielded a veritable explosion of new insights, with two Fields Medals being awarded to Smirnov and Werner for their breakthroughs.

In this course, we aim to familiarize the audience with a modern approach to some classical results from the probabilistic theory of lattice models, using Bernoulli percolation and the Random-Cluster model as our main examples. We then use these tools to discuss some very recent results on the study of random Lipschitz functions.

A particular focus will be given to the two-dimensional models, where even the simplest models lead to a dazzling array of different fractal behaviors. This is a consequence of the conformal invariance of these models, which is predicted for all the models discussed, but rigorously proved in very few cases. One of our goals is a presentation of Smirnov's proof of the conformal invariance of critical site percolation on the triangular lattice.

These models give a beautiful way to apply the material learnt in the Probability course. Quite a few ideas are of combinatorial nature and the field is connected to several other branches of mathematics: Mathematical Physics, Ergodic Theory, Complex Analysis, Conformal Geometry, Computer Science.

It is highly recommended to follow the exercise sessions.

Assessment and permitted materials

oral exam (possibly online)

Minimum requirements and assessment criteria

Thorough understanding and a working knowledge of the core part of the material presented in the lectures is required for passing the exam.

Examination topics

The material presented in the lectures; a more detailed description of what exactly is expected in the exam will be made available during the lectures

Reading list

Lecture notes of Hugo Duminil-Copin:
- https://www.ihes.fr/~duminil/publi/2017percolation.pdf
- https://arxiv.org/pdf/1707.00520.pdf

Grimmett "The Random-cluster model":
- http://www.statslab.cam.ac.uk/~grg/books/rcm1-1.pdf

Association in the course directory

MSTV

Last modified: We 29.03.2023 09:28