Advanced Topics in Probability (0366-4374)
Fall 2021, Tel Aviv University
Location: Dan-David Building 106, Wednesdays 13:10 - 16:00. Lectures are recorded and students may also join the classes remotely via the Zoom link.
Instructor: Ron Peled.
Prerequisites: The course Probability for Mathematicians.
Grade: The final grade will be based on the solution of homework problems.
Statistical physics is the branch of physics studying how the large-scale properties of systems emerge from the local interactions of the microscopic particles of the system. The mathematical study of statistical physics has formed a central part of probability theory research in the last century and its importance has further been emphasized by many breakthroughs in the last 20 years. Research is driven by detailed investigations of specific models, such as random walks (or Brownian motion), the Ising and spin O(n) models, percolation and many others.
The course will serve as a gentle introduction to some problems at the forefront of current research in statistical physics. No prior knowledge of physics or statistical physics models will be assumed.
The general theme of the course will be the effect of disorder on statistical physics models. By this we mean the way in which the behavior of specific models is altered when they are placed in an inhomogeneous (typically random) environment. This theme will provide an opportunity to describe both the `pure' models (i.e., the models in an homogeneous environment) and to introduce the conjectures, open problems and results in the study of disorder effects. Along the way we will learn several techniques of general interest in probability theory and describe their application to the specific problems.
Notices: The last class of the semester, on January 5, will be held by Zoom (see above link) due to the rising pandemic levels.
Suggested Topics
Among the suggested topics for the course (we will only cover a selection of these) are the study of
- Random Walks in Random Environment.
- Disordered Spin Systems such as the Random-Field Ising and Spin O(n) Models and Spin-Glass Models.
- Noise Sensitivity and Dynamical Percolation.
- Random Operators and Anderson Localization.
These topics will be discussed on the hypercubic lattice Zd.
Relevant Books and Lecture Notes
Random walks in random environment
Lattice spin systems
Noise Sensitivity and Dynamical Percolation
Disordered quantum spin systems
Homework
The homework problems are available here. Problems will be added on a continuous basis during the semester. Please submit homework in English, if possible.
The final grade will be based on the solution of the homework problems. In order to obtain a full grade, 8 homework problems should be solved correctly. Among the 8 problems, it is required that at least one and at most three problems be solved from each of the four sections in the homework file. Students are allowed to discuss the problems between them but the writing of solutions should be done individually by each student.
The homework should be submitted to the instructor's (physical or digital) mail box by February 6. Write an email to notify the instructor if a physical copy is submitted.
Lectures
- Lecture 1 (13.10): Introduction to the course (Statistical physics. Review of specific models. The idea of adding disorder to a model). Overview of random walks on Zd in a homogeneous environment, including a proof of the central limit theorem via Lindeberg's method.
Lecture notes scribed by Nick Kushnir.
- Lecture 2 (20.10): General definitions on nearest-neighbor random walk in random environment on Zd (environment measure, ergodicity and uniform ellipticity, quenched and annealed walk measures). Random walk in random environment in one dimension: Statement of Solomon's theorems on recurrence/transience and law of large numbers, proof of the former and beginning of proof of the latter. A brief description of Birkhoff's ergodic theorem.
Based on the lecture notes of Ofer Zeitouni.
Lecture notes scribed by Maya Rat.
- Lecture 3 (27.10): The law of large numbers for random walk in random environment on Z. Discussion of branching process representation.
Based on the lecture notes of Ofer Zeitouni.
Lecture notes scribed by Nick Kushnir.
- Lecture 4 (3.11): Limit theorems for random walk in transient random environment on Z (using a central limit theorem for partial sums of a stationary sequence). Discussion of the slowdown created by traps (using Cramér's theorem on large deviations for partial sums of an IID sequence). Discussion of the recurrent case. Brief introduction to random walk in random environment on Zd for d≥2.
Based on the lecture notes of Ofer Zeitouni.
Lecture notes scribed by Nick Kushnir.
- Lecture 5 (10.11): Random walk in random environment on Zd for d≥2: Kalikow's 0-1 law and sketch of the Merkl-Zerner 0-1 law in two dimensions.
Based on the lecture notes of Ofer Zeitouni.
Lecture notes scribed by Maya Rat.
- Lecture 6 (17.11): Random walk in random environment on Zd for d≥2: The Merkl-Zerner counterexample to the 0-1 law in stationary and ergodic environments. Discussion of related results, questions and conjectures. Introduction to Kalikow's condition for transience and ballisticity.
Based on the lecture notes of Ofer Zeitouni and the lecture notes of Drewitz and Ramirez.
Lecture notes scribed by Maya Rat.
- Lecture 7 (24.11): Random walk in random environment on Zd for d≥2: End of discussion of Kalikow's condition. Sznitman's (T) condition and polynomial extensions. Brief discussion of the environment viewed from the point of view of the particle.
Based on the lecture notes of Ofer Zeitouni and the lecture notes of Drewitz and Ramirez.
Lecture notes scribed by Nick Kushnir.
- Lecture 8 (1.12): Brief discussion of special cases of random walk in random environment on Zd (d≥2) for which more is known: Balanced random environment, small perturbations of simple random walk, simple random walk in a subset of coordinates, random conductance model (see the lecture notes of Marek Biskup) and Dirichlet random environment.
Second course topic: Lattice spin systems.
An introduction to statistical physics and lattice spin systems. Examples: Ising and spin O(n) models. Potts model. Overview of the phase transitions associated with the magnetization in the Ising and spin O(n) models according to the spatial dimension d and the number of spin components n.
Random walk discussion based on the lecture notes of Ofer Zeitouni, the lecture notes of Drewitz and Ramirez and the lecture notes of Bogachev.
Spin systems discussion based on the lecture notes of Peled–Spinka.
Lecture notes scribed by Nick Kushnir.
- Lecture 9 (8.12): Dobrushin uniqueness condition - a general method for proving exponential decay of correlations in spin systems. Overview (framework, spatial mixing, Wasserstein distance, statement) and proof.
Based on Lecture 1 of the random proper colorings lecture notes of Peled–Spinka and self-notes (for the extension from total variation distance to Wasserstein distance).
Lecture notes scribed by Nick Kushnir.
- Lecture 10 (15.12): Long-range order in the Ising model in dimensions d≥2 - the Peierls argument. Part of the proof of absence of long-range order in the two-dimensional spin O(n) model with n≥2 - the Mermin-Wagner theorem.
Based on the lecture notes of Peled–Spinka and self-notes.
Lecture notes scribed by Nick Kushnir.
- Lecture 11 (22.12): End of proof of the Mermin-Wagner theorem. Long-range order in the d≥3 spin O(n) model with n≥2 - the infra-red bound (proof of long-range order from Gaussian domination).
Based on self-notes and the Marseille lectures of Daniel Ueltschi.
Lecture notes scribed by Nick Kushnir.
- Lecture 12 (29.12): End of the proof of long-range order in the d≥3 spin O(n) model with n≥2 (proof of Gaussian domination using reflection positivity).
Disordered lattice spin systems: Examples (random-field Ising, Potts and spin O(n) models. Disordered ferromagnet and spin glasses). Introduction to the Imry-Ma phenomenon and results. Further information on the random-field Ising model. Conjecture of unique ground state in d=2 random-field Potts model and unique ground-state pair in d=2 spin glass model.
Based on the Marseille lectures of Daniel Ueltschi and on self-notes.
Lecture notes scribed by Roey Zemmel.
- Lecture 13 (5.1): Random-field Ising and Potts models: A quantitative bound on the rate of correlation decay in two dimensions. Overview of the Ding-Zhuang proof of long-range order in dimensions d≥3.
Based on the paper (see also Slides 1, Slides 2 and video), on the paper and self-notes.
Lecture notes delivered during online class.