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 – Technion graduate course blog (2018/19)
Random Graphs – general topics and older posts
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 in Oxford on Evan’s Probability and Real Trees
January-March 2015. These are based on the sessions various delivered by members of the group:
Combinatorial Stochastic Processes
Posts based on the book by Jim Pitman following his 2002 Saint-Flour course. 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:
The discrete Gaussian free field
The DGFF was the main object of study in my postdoc project at Technion. I found aspects of this topic hard to pick up as rapidly as I’d hoped, so perhaps the following posts might be helpful to anyone in a similar position in the future!
- The DGFF from scratch
- Boundary conditions and the Gibbs-Markov property
- Gibbs-Markov for entropic repulsion
- Properties of the Green’s function
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.
- Motivation and Cramer’s Theorem
- LDPs, Rate Functions and Lower Semi-Continuity
- Gartner-Ellis: where do all the terms come from?
- Sanov’s Theorem
- Stochastic Processes and Mogulskii’s Theorem
- LDPs for Random Graphs
- Azuma-Hoeffding Inequality
Several other posts reference and use this ideas. Searching for Large Deviations
will reveal these.
Markov Chains and Mixing Times
Posts based on a reading group in Oxford 2012/13, 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.
- Reversing Markov Chains
- Metropolis Chains
- Convex Functions on the Space of Measures
- Avoiding Periodicity
- Cesaro Mixing
- The Aldous-Broder Algorithm and Cover Times
- Mixing of the Noisy Voter Model
- 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
These are a handful 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:
These posts were specifically motivated by courses I taught in Oxford. For comparison with other universities, 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