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Modern Discrete Probability: An Essential Toolkit

(To be published by Cambridge University Press)

Sebastien Roch, Department of Mathematics, UW-Madison

Course Information for Fall 2023

In Fall 2023, MATH/STAT 833 will cover selected topics from the notes below. All announcements for the course will be made on Piazza, which you can also access through Canvas.

Note: If you are currently on the waitlist, you can view the latest announcements on the public Piazza course page.


The goal of these notes is to give an introduction to fundamental models and techniques in graduate-level modern discrete probability. Topics are taken mostly from probability on graphs: percolation, random graphs, Markov random fields, random walks on graphs, etc. No attempt is made at covering these areas in depth. Rather the emphasis is on developing and illustrating common and important techniques. Various applications, in particular in the theoretical foundations of data science and machine learning, are also discussed along the way.

The notes are aimed at graduate students in mathematics, statistics, computer science, electrical engineering, physics, economics, etc. with previous exposure to basic probability theory (ideally measure-theoretic probability theory; at Wisconsin, Math 733; my own course notes) and stochastic processes (at Wisconsin, Math 632). These notes were developed for a one-semester course at UW-Madison (Fall 2014, Fall 2017, Fall 2020, and Fall 2023). I also gave a version of this course at the TRIPODS Summer School on Fundamentals of Data Analysis. The slides are here.

Slides for background sections are also available below.

Much of the material covered can also be found in the following excellent texts (conveniently available online):

Lectures Notes

Full notes in one file (12MB) (updated: mar 6, 2023)

Front matter (table of contents, preface, notation) (updated: mar 6, 2023)

Chapter 1: Introduction (updated: mar 6, 2023)

Chapter 2: Moments and tails (updated: mar 6, 2023)

Chapter 3: Martingales and potentials (updated: mar 6, 2023)

Chapter 4: Coupling (updated: mar 6, 2023)

Chapter 5: Spectral methods (updated: mar 6, 2023)

Chapter 6: Branching processes (updated: mar 6, 2023)

Back matter (appendices, bibliography, index) (updated: mar 6, 2023)


Slides covering mostly background material (and more) in each chapter:

Last updated: aug 29, 2023.