LFU:online

On completion of this couse, students have acquired a deep knowledge of methods used in lattice models. Moreover they are familiar with relevant mathematical literature and can judge its mathematical content. They are able to examine problems related to random walks and matrices in a creative and methodically correct manner and to present the result of those examinations in written form and orally as to be understood well by experts.

This research seminar offers the opportunity to present one's own research results, or explore and present a current topic from lattice models. We will concentrate on the classical book of Grimmett "Random-cluster model" as well as on more recent results in lattice models.

This is a very active of research in probability theory, with connections to mathematical physics and complex analysis. The growing interest to lattice models is confirmed with Fields Medals to Werner (2006), Smirnov (2010), Duminil-Copin (2022). Berezinskii–Kosterlitz–Thouless phase transition in the XY model was awarded Nobel Prize in Physics in 2016.

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.

The topic is related to VU Percolation theory and applications, but there is no direct depency: one can follow follow VU, SE or both.

Disscussions and presentations of parts of the book of Grimmett "Random-cluster model" as well as more recent results in lattice models.

Continuous assessment (based on regular written and oral contributions by the participants).

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

Duminil-Copin "Lectures on the Ising and Potts models on the hypercubic
lattice" https://arxiv.org/pdf/1707.00520

The topic is related to VU Percolation theory and applications, but there is no direct depency: one can follow follow VU, SE or both.

The topic is related to VU Percolation theory and applications, but there is no direct depency: one can follow follow VU, SE or both.