Contents

It seems very impractical to attempt to read a blog in any manner other than chronologically. However, while it is very appealing to the author to flit between topics, this may be rather annoying for the reader who is looking for something more specific. I hope the following vague list of themes and posts is helpful.

Random Graphs

Random Trees and Random Maps

Based on the course on Random Maps given by Gregory Miermont in Saint-Flour 2014:

Matthias Winkel ran a reading group on Evan’s Probability and Real Trees January-March 2015. These are based on the sessions various members of the group gave:

Combinatorial Stochastic Processes

I’ve been reading the book by Jim Pitman based on his notes from St. Flour in 2002. The book is available from Springer, and online here. There are ten chapters which are slightly related on a range of interesting topics, some of which overlap with other things in this list. Some of the following posts are explicitly motivated by this:

Large Deviations

This series is motivated by a course I took through the Taught Course Centre (via video link from Warwick) in 2012. Ideas for posts 1-5 are drawn mainly from den Hollander and Dembo/Zeitouni’s books.

  1. Motivation and Cramer’s Theorem
  2. LDPs, Rate Functions and Lower Semi-Continuity
  3. Gartner-Ellis: where do all the terms come from?
  4. Sanov’s Theorem
  5. Stochastic Processes and Mogulskii’s Theorem
  6. LDPs for Random Graphs
  7. Azuma-Hoeffding Inequality

Several other posts reference and use this ideas. Searching for Large Deviations will reveal these.

Random Walks

Markov Chains and Mixing Times

Three of us started a reading group devoted to this book by Levin, Peres and Wilmer. The text is available online here. There is roughly one post per two chapters for the first 12 chapters, which is the ‘core’ material in some sense.

  1. Reversing Markov Chains
  2. Metropolis Chains
  3. Convex Functions on the Space of Measures
  4. Avoiding Periodicity
  5. Cesaro Mixing
  6. The Aldous-Broder Algorithm and Cover Times
  7. Mixing of the Noisy Voter Model
  8. Coupling from the Past

A more accessible discussion of the Top-to-Random shuffle in three parts begins here and continues with Part II and Part III.

Combinatorics and non-Random Graph Theory

Percolation

Networks

These are three posts about real-world networks

I also wrote some revision posts on the Cambridge Part III course Stochastic Networks. The topics include: queues, Braess’ paradox, random access, effective bandwidth and loss networks. Start here and click next up to five times!

Stochastic Calculus and General Probability

There are loads of posts on this from the Part III courses Advanced Probability and Stochastic Calculus among others. The highlights include:

For some reason, the following are among the most popular (or at least, the most likely to come up on Google…) articles I’ve written:

More recent posts on these topics, based on machinery I’ve needed in research:

Teaching

These posts were specifically motivated by courses I have been involved in teaching. For comparison with other universities which use a more conventional indexing than Oxford, note Prelims = first year, Part A = second year, Part B = third year.

I also taught a lecture course on Markov Chains in China in August 2012. I wrote a diary about the trip. You can find the first part here, and click next for the remaining parts. I also wrote two posts about Poisson Processes, and one about invariant distributions.

Posts about material and travel related to olympiads are linked from this page.

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