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Modern Discrete Probability: An Essential Toolkit

(Lecture notes)

Sebastien Roch, UW-Madison


The goal of these notes is to give an introduction to fundamental models and techniques in graduate-level modern discrete probability. Topics covered 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 illustrating common and important techniques.

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 and Fall 2017).

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

Table of contents (last updated: sep 9, 2015)

Chapter 1: Some important models (draft last updated: sep 9, 2015)

Chapter 2: Moments and tails (draft last updated: sep 9, 2015)

Chapter 3: Martingales and potentials (draft last updated: sep 9, 2015)

Chapter 4: Coupling (draft last updated: sep 9, 2015)

Chapter 5: Branching processes (draft last updated: sep 9, 2015)

Chapter 6: Eigenvalues and isoperimetry (in preparation)

Chapter 7: Influence (planned)

Appendix, bibliography and index (last updated: sep 9, 2015)


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

Last updated: sep 6, 2017.