Ma/ACM/IDS 140c - Spring 2025

Ma/ACM/IDS 140c: Probability (Spring 2025)


Instructor Lingfu Zhang (lingfuz@caltech Office Linde Hall 358)

Class time 10:30-11:50 on Tuesdays and Thursdays
Class location Linde Hall 183
Office hour TBD

Course description


This course aims to survey fundamental concepts and selected topics in modern probability theory, with a focus on asymptotic behavior. We will begin by introducing key tools such as the moment method and concentration inequalities, with applications in e.g., random graphs. The course will then move into random matrix theory, covering classical models and techniques related to the eigenvalues of random matrices, including the Wigner semicircle law, \(\beta\)-ensemble statistics, determinantal point processes, and bulk and edge limits. The second direction is interacting particle systems, where we will discuss models such as the voter model, exclusion process, and contact process.

The course is intended for advanced undergraduate students and PhD students from mathematics and related departments.

Grading For students who need to receive a grade for this course, there are three problem sets, which may be done in collaboration. There will be no exam.

Reference material

  • Modern Discrete Probability: An Essential Toolkit by Sebastien Roch.
  • Random Graphs and Complex Networks by Remco van der Hofstad.
  • Random Graph Dynamics by Rick Durrett.
  • Large Deviations Techniques and Applications by Amir Dembo and Ofer Zeitouni.
  • Topics in random matrix theory by Terrence Tao.
  • An Introduction to Random Matrices by Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni.
  • Spectral Analysis of Large Dimensional Random Matrices by Zhidong Bai and Jack W. Silverstein.
  • Interacting Particle Systems by Thomas M. Liggett.
  • Probability on Graphs by Geoffrey Grimmett.
  • Reversible Markov Chains and Random Walks on Graphs by David Aldous and James A. Fill.
  • Integrable stochastic interacting systems by Tomohiro Sasamoto.

    • Schedule

    Apr 1 Introduction. Moment methods, random permutation, Erdős–Rényi graph. (Roch 2.1, 2.2)
    Apr 3 Random \(d\)-regular graph and configuration model, connectivity of random graphs. (Roch 2.2, van der Hofstad 7.2, 7.4-7.5, Durrett Chapter 3)
    (Notes for Apr 1 and Apr 3.)
    Apr 8 Concentration inequalities, large deviation, and applications. (Roch 2.4, 3.2, DZ 2.2)
    (Notes for Apr 8.)
    Apr 10 Monotone coupling and FKG. (Roch 4.2)
    (Notes for Apr 10.)

    Random Matrix Theory

    Apr 15 Wigner matrices, convergence of spectrum, moment method. (Tao 2.3, 2.4, AGZ 2.1)
    Apr 17 Semi-circle law and operator norm via moments, Stieltjes transform. (Tao 2.3, 2.4, AGZ 2.1, 2.4)
    Apr 22 Semi-circle law via Stieltjes, Marchenko-Pastur law. (AGZ 2.4, BS Chapter 3)
    (Notes for Apr 15, 17, 22.)
    Apr 24 Gaussian/Wishart/Jacobi Unitary/Orthogonal matrices. (AGZ 2.5, 4.1)
    Apr 29 Bulk and edge limits, determinantal point process. (AGZ 3.1, 3.4, 4.2)
    May 1 Sine and Airy kernels. (Tao 3.2, AGZ 3.5, 3.7)
    May 6 Tracy-Widom distributions, tri-diagonal matrices. (AGZ 3.8, 4.5)
    May 8 Matrix addition and free convolution. (Tao 2.5)

    Interacting Particle System

    May 13 Duality, Voter model, coalescing random walks. (Liggett 2.3, 5.1-5.3, AF 14.3)
    May 15 Contact process (SIS). (Grimmett 6.3-6.6, Liggett Chapter 6)
    May 20 Exclusion process, coupling and invariant measures. (Liggett 8.1, 8.2)
    May 22 Asymmetric simple exclusion process, Bethe ansatz. (Sasamoto 1.1-1.4)
    May 27 Multi-species exclusion process, Hecke algebra, symmetry.
    May 29 Queueing, Burke's theorem, multi-species invariant measure.

    Problem sets

    To be released.

    Discussion of the problem and how to solve it is allowed. However, intermediate work and the final solution should not be shared.