Around the Ising Model in 80 days I
I will discuss the simplest model of Statistical Mechanics: the 2D Ising model, a drosophila of mathematical physics. Yet there are open problems left and many papers are still written on it. I will also treat related models: thе independent percolation, the models with continuous symmetry, the models on other lattices, like the trees and the Lobachevsky plane.
Keywords: phase transition, criticality, correlation decay, Gibbs state, Markov random field, conformal invariance,… All will be explained in full details.
Keywords: phase transition, criticality, correlation decay, Gibbs state, Markov random field, conformal invariance,… All will be explained in full details.
Lecturer
Date
4th November, 2022 ~ 13th January, 2023
Website
Prerequisite
Undergraduate Probability: Independent events, Laws of large numbers. Basic measure theory and functional analysis.
Syllabus
1. Independent percolation
Phase transition: the appearance of the infinite cluster.
Coupling and monotonicity. Kantorovich distance.
Ergodicity and the number of infinite clusters.
Pivotal edges and the Margulis-Russo identity.
Kesten: critical p=1/2 for Z^2. FKG inequality.
Russo-Seymour-Welsh theory.
Discrete analytic functions. Conformal invariance of the 2D percolation.
2. Ising model
Markov chains and Gibbs states: the problems of existence and uniqueness.
Ground states, their stability.
Constructive uniqueness.
Phase diagrams of the ferromagnetic and antiferromagnetic Ising models.
Surface tension. Interfaces, and the roughening phenomenon.
Potts model and its critical point.
3. Models with continuous symmetry
Mermin-Wagner theorem.
Reflection Positivity.
The Berezinskii–Kosterlitz–Thouless transition.
Phase transition: the appearance of the infinite cluster.
Coupling and monotonicity. Kantorovich distance.
Ergodicity and the number of infinite clusters.
Pivotal edges and the Margulis-Russo identity.
Kesten: critical p=1/2 for Z^2. FKG inequality.
Russo-Seymour-Welsh theory.
Discrete analytic functions. Conformal invariance of the 2D percolation.
2. Ising model
Markov chains and Gibbs states: the problems of existence and uniqueness.
Ground states, their stability.
Constructive uniqueness.
Phase diagrams of the ferromagnetic and antiferromagnetic Ising models.
Surface tension. Interfaces, and the roughening phenomenon.
Potts model and its critical point.
3. Models with continuous symmetry
Mermin-Wagner theorem.
Reflection Positivity.
The Berezinskii–Kosterlitz–Thouless transition.
Audience
Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Senya Shlosman obtained his PhD in 1978, from the St.-Petersburg branch of Steklov institute. His adviser was Roland Dobrushin. He obtained his second PhD (habilitation) in 1989, from the Ukrainian Institute of Mathematics. From 1991 he was Professor of the Dept. of Math., UC Irvine. He moved to France in 1996, getting the position of Directeur de Recherche in CPT, CNRS, Luminy, Marseille. Currently he is a leading scientific researcher in the Institute for Information Transmission Problems of the Academy of Science, Moscow, and Professor of the Center for Advance Studies in Skolkovo Inst. of Technology, Moscow.