For some models there is a such that the global behaviour of the system is quite different for and for. Clearly and, since there are no open edges at all when and all edges are open when. 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 ?' A limit as of the question raised above is `What is the probability that there exists an open path from to infinity?' This probability is called the percolation probability and is denoted by. All edges of are, independently of each other, chosen to be open with probability and closed with probability. The simplest version of percolation takes place on, which we view as a graph with edges between neighbouring vertices. ![]() Percolation is a simple probabilistic model which exhibits a phase transition. Let us briefly explain the mathematical setting using the example of Bernoulli percolation. Leads to: The following modules have this module listed as assumed knowledge or useful background: Synergies: The following modules go well together with Markov Processes: ![]() 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. Thus the course will equip you with modern tools for studying probabilistic models of phase transitions. This is in part due to simplified proofs due to Duminil-Copin, Hofstadt, Heydenreich and many others which appeared only in the last decade. The beauty of the models we are studying in the course is in the possibility to understand them using elementary probabilistic methods. the continuity of the percolation function), which will certainly bring you the Fields medal if you can answer them before you are 40! Yet there are still some fundamental unresolved questions (e, g. For example, there have been already two Fields medals awarded in the 21st century for studying percolations (Smirnov and Duminil-Copin). 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. Probabilistically it is the simplest model of spatial disorder. Bernoulli percolations were introduced in the late 1950's to model the propagation of fluid through porous media and gained. 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. ![]() 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. magnetisation) on the parameters controlling the phase (e. ![]() One source of difficulty is the non-analytical dependence of the observables detecting the phase transition (e. However the rigorous mathematical theory of phase transition is both exciting and hard. Phase transitions are ubiquitous in Nature: freezing and evaporation of water and spontaneous magnetisation of a ferromagnet are some of the most familiar examples. Useful background: This module provides an introduction to phase transitions for Markov processes and Bernoulli percolation models. 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'. 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. Assumed knowledge: MA359 Measure Theory and ST342 Mathematics of Random Events.
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