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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). 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):

- [vdH] Random Graphs and Complex Networks. Vol. I by van der Hofstad
- [LP] Probability on Trees and Networks by Lyons with Peres
- [LPW] Markov Chains and Mixing Times by Levin, Peres and Wilmer
- [St] A Mini Course on Percolation Theory by Steif
- [Gr] Probability on Graphs by Grimmett
- [Lu] Concentration-of-measure Inequalities by Lugosi
- [Ve] High-dimensional probability: An introduction with applications in data science by Vershynin

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

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

**Review:**basics of graph theory and Markov chain theory.**Models:**percolation, random graphs, Markov random fields, random walks on networks, interacting particles on finite graphs.

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

**Techniques:**first moment method, second moment method, Chernoff-Cramer method, matrix tail bounds (planned).**Examples:**percolation on trees and lattices (critical value), Erdos-Renyi graph (containment, connectivity), Markov chains (Varopoulos-Carne), random k-SAT threshold, Johnson-Lindenstrauss.

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

**Review:**stopping times, martingales.**Techniques:**concentration for martingales, method of bounded differences, electrical networks.**Examples:**Markov chains (hitting times, cover times, recurrence), percolation on trees (critical regime), Erdos-Renyi graphs (chromatic number), preferential attachment graphs (degree sequence), uniform spanning trees (Wilson's method).

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

**Review:**definition of coupling.**Techniques:**coupling inequality, stochastic domination, correlation inequalities (FKG, Holley), couplings of Markov chains.**Examples:**harmonic functions on lattices and trees, Markov chains (mixing time, Glauber dynamics), Erdos-Renyi graph (degree sequence, Janson's inequality), percolation on lattices (RSW theory, Harris' theorem), and Ising model on lattices (extremal measures).

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

**Review:**Galton-Watson processes, extinction.**Techniques:**Random-walk representation, duality principle, comparison to branching processes.**Examples:**percolation on trees (critical exponents) and on Galton-Watson trees, Erdos-Renyi graph (phase transition).

*Chapter 6:* Eigenvalues and isoperimetry (in preparation)

**Techniques:**spectral gap, canonical paths, bottleneck ratio, expander graphs, Talagrand's inequality.**Examples:**Markov chains (mixing time, Glauber dynamics), percolation on lattices (uniqueness of infinite cluster) and on Galton-Watson trees, stochastic traveling salesman.

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

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

- Slides for Chapter 1 (last updated: sep 6, 2017)
- Slides for Chapter 3 (last updated: jan 2, 2015)
- Slides for Chapter 4 (last updated: nov 29, 2014)
- Slides for Chapter 5 (last updated: nov 15, 2014)
- Slides for Chapter 6 (last updated: dec 22, 2014)

Last updated: oct 26, 2018.