# MA3H2 Markov Processes and Percolation Theory

Lecturer: Oleg Zaboronski

Term(s): Term 2

Status for Mathematics students: List A

Commitment: 30 lectures

Assessment: 3 hour exam 100%

Formal registration prerequisites: None

Assumed knowledge: MA359 Measure Theory and  ST342 Mathematics of Random Events. Alternatively, the students need to know the following basic facts: probability measure and expectation (including conditional expectation); convergence of random variables; the law of large numbers and central limit theorems; basic theory of Markov chains and random walks; relevant theorems of analysis such the Fubini's theorem, the dominated and the monotone convergence theorems. Most of the above facts are summarised on the course's Moodle page and covered by Chapter 1 and the Appendix of Rick Durret's book 'Probability: theory and examples'.

Useful background: This module provides an introduction to phase transitions for Markov processes and Bernoulli percolation models. Phase transitions are ubiquitous in Nature: freezing and evaporation of water and spontaneous magnetisation of a ferromagnet are some of the most familiar examples. However the rigorous mathematical theory of phase transition is both exciting and hard. One source of difficulty is the non-analytical dependence of the observables detecting the phase transition (e. g. magnetisation) on the parameters controlling the phase (e. g. temperature). In the course we will treat rigorously two of the simplest models exhibiting phase transition: firstly, we will investigate the extinction phase transition for the well know Galton-Watson branching process from population dynamics, secondly - the percolation transition for Bernoulli percolation model on tree graphs and $Z to the d$.

Galton-Watson branching process was introduced in the 19th century to investigate the chance of the perpetual survival of aristocratic families in Victorian Britain and has since became both a useful model for population dynamics and an interesting probabilistic model in its own right. Bernoulli percolations were introduced in the late 1950's to model the propagation of fluid through porous media and gained. Probabilistically it is the simplest model of spatial disorder. Each of the models is very easy to define, yet there are still many open research questions concerning both the branching process and the percolation model. For example, there have been already two Fields medals awarded in the 21st century for studying percolations (Smirnov and Duminil-Copin). Yet there are still some fundamental unresolved questions (e, g. the continuity of the percolation function), which will certainly bring you the Fields medal if you can answer them before you are 40!

The beauty of the models we are studying in the course is in the possibility to understand them using elementary probabilistic methods. This is in part due to simplified proofs due to Duminil-Copin, Hofstadt, Heydenreich and many others which appeared only in the last decade. Thus the course will equip you with modern tools for studying probabilistic models of phase transitions. The acquired knowledge will allow you understand research papers on branching processes and percolations and will be applicable to the study phase transitions in applications such as biological and physical systems, communication networks and financial markets.

Synergies: The following modules go well together with Markov Processes:

Leads to: The following modules have this module listed as assumed knowledge or useful background:

Content:

Let us briefly explain the mathematical setting using the example of Bernoulli percolation. Percolation is a simple probabilistic model which exhibits a phase transition. The simplest version of percolation takes place on $\mathbb{Z} ^2$ , which we view as a graph with edges between neighbouring vertices. All edges of $\mathbb{Z}^2$ are, independently of each other, chosen to be open with probability $p$ and closed with probability $1-p$ . A basic question in this model is What is the probability that there exists an open path from the origin to the exterior of the square $S_n=[-n,n]^2$?' A limit as $n\to\infty$of the question raised above is What is the probability that there exists an open path from $0$to infinity?' This probability is called the percolation probability and is denoted by $\theta(p)$ . Clearly $\theta(0)=0$ and $\theta(1)=1$, since there are no open edges at all when $p=0$ and all edges are open when $p=1$ . For some models there is a $0<p_{c}<1$ such that the global behaviour of the system is quite different for $p<p_{c}$ and for $p>p_{c}$. Such a sharp transition in global behaviour of a system at some parameter value is called a phase transition or a critical phenomenon, and the parameter value at which the transition takes place is called a critical value.

Books:

We will not follow a particular book. However, there are several sets of lecture notes used in the course, which can be downloaded from the Moodle page. The list below is a selection of books for a much deeper study of the subject.

H.O. Georgii: Stochastics: introduction to probability theory and statistics, de Gruyter (2008). [basic introduction to stochastics and Markov chains (discrete time)]

J. Norris: Markov chains, Cambridge University Press [standard reference treating the topic with mathematical rigor and clarity, and emphasizing numerous applications to a wide range of subjects]

G. Grimmett, D. Stirzaker: Probability and Random Processes, OUP Oxford (2001) [chapter 6 on Markov chains]

G. Grimmett: Probability on Graphs, Cambridge University Press (2010). [Available Online, contains a nice introduction to processes on graphs and percolation]

B. Bollabás, O. Riordan: Percolation, Cambridge University Press (2006). [a modern treatment of percolation. The introduction and the chapter on basic techniques are relevant for the lecture]

G. Grimmett: Percolation, 2nd ed., Springer (1999). [the standard reference on percolation. It contains much more than covered in the lecture. The first two chapters are relevant for the lecture]