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 – Technion graduate course blog (2018/19)

• Lecture 1: Introduction to random graphs
• Lecture 2: Connectivity threshold for G(n,p)
• Lecture 3: Coupling, comparing distributions
• Lecture 4: Exploration processes, and the hitting time theorem
• Lecture 6: Local limits
• Lecture 7: The giant component
• Lecture 8: Component sizes in the critical window
• Lecture 9: Inhomogeneous random graphs
• Lecture 10: The configuration model

Random Graphs – general topics and older posts

• Exploring the Supercritical Random Graph
• Analytic vs Probabilistic Arguments for a Supercritical Branching Process
• G(n,p) vs G(n,m)
• LDPs for Random Graphs
• Long Paths and Expanders
• Geometric Random Graphs
• Critical Components in Erdos-Renyi
• Isolated Vertices and the Second-Moment Method
• Random 3-regular graphs
• Kernels of critical graph components

Random Trees and Random Maps

• Uniform Spanning Trees
• Minimum Spanning Trees
• Generating Uniform Trees
• Diameters of Trees and Cycle Deletion
• Bijections, Prufer Codes and Cayley’s formula
• Local Limits
• Discontinuous Phase Transitions
• Multitype Branching Processes

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

• Towards the Schaeffer Bijection
• The Schaeffer Bijection
• Leaves and Geodesics in BCRT

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:

• Gromov-Hausdorff Distance and Correspondences
• Gromov-Hausdorff Distance on Trees
• Real Trees – Root Growth and Regrafting

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:

• Bell Polynomials
• Enumerating Forests
• Increments of Random Partitions
• Urns and the Dirichlet Distribution
• The Contour Process
• The Chinese Restaurant Process
• Random Mappings for Cycle Deletion
• The Coupon Collector Problem

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!

1. The DGFF from scratch
2. Boundary conditions and the Gibbs-Markov property
3. Gibbs-Markov for entropic repulsion
4. Properties of the Green’s function

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

• Random Walks and Spanning Trees
• Branching Random Walk and Amenability
• Lamperti Walks
• Skorohod embedding
• Random walks conditioned to stay positive
• Trap models and laws of not-so-large numbers

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.

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

• Long Paths and Expanders
• Hall’s Marriage Theorem
• Generating Functions
• Rhombus Tilings
• Combinatorial Nullstellensatz
• Point Set Combinatorics
• Lovasz Local Lemma
• Turan’s Theorem
• Chains and antichains
• Antichains in the grid

Percolation

• Independence and Association
• Random Interlacements
• Characterisations of Geometric Random Graphs
• Discontinuous Phase Transitions
• Noise Sensitivity and Influence
• Sharpness of Phase Transition in Percolation
• Parking on a ring, linear hashing

Networks These are a handful posts about real-world networks

• Beyond Erdos-Renyi
• Preferential Attachment Models
• The Configuration Model
• Persistent Hubs
• Dispersion in Social 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:

• Motivating Ito’s formula
• Remarkable fact about BM 1, and three more.
• Recurrence and Transience of BM
• Towards the Ito Isometry

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:

• Brownian Excursions and Local Time
• The Levy-Khintchine Formula
• Skorohod Space

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

• Reflected Brownian Motion
• Sticky Brownian Motion
• Ornstein-Uhlenbeck Process
• Skorohod Representation Theorem
• Non-separable Skorohod Representations
• Tightness in Skorohod Space
• Doob inequalities and Doob-Meyer decomposition

Teaching 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.

• Part B Applied Probability: DBEs; Inspection Paradox, The Yule Process
• Part A Probability: Convergence of Transition Probabilities, Avoiding Mistakes in Probability Exams, Hitting Probabilities for Markov Chains
• Part A Statistics: Extreme Value Theory, Bayesian Inference and the Jeffreys Prior, Fisher Information and Cramer-Rao, Questionable Statistics
• Prelims Linear Algebra II: Determinants, Determinants continued, Eigenvectors and diagonalisability

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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