
Academic Year:  2012/3 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 
Level:  Masters UG & PG (FHEQ level 7) 
Period: 
Semester 1 Semester 2 
Assessment:  EX 100% 
Supplementary Assessment:  Mandatory extra work (where allowed by programme regulations) 
Requisites:  Before taking this unit you must take MA40125 or you must have taken MA20225 and have consulted the unit lecturer. 
Description:  Aims: To introduce some discrete probabilistic structures with relevance to networks, statistical physics and communications  in particular, random graphs and percolation. To present some of the key results on these structures and the mathematical ideas and techniques behind these results. To acquaint the student with notions of phase transition, spatial and combinatorial methods in probability theory. Learning Outcomes: On completing the course, students should be able to: * Describe the giant component phenomenon in random graphs and the phase transition phenomenon in percolation theory; * Compute percolation thresholds on graphs such as trees; * Perform simple computations on discrete probability structures (for example, mean subgraph counts on random graphs, or lattice animal counts); * Apply tools such as Harris' inequality and branching process approximation; * Appreciate ideas behind such theorems as the infinite cluster's uniqueness. Skills: Numeracy T/F A Problem Solving T/F A Written and Spoken Communication F (in tutorials) Content: Random Graphs: definition and motivation; small subgraphs; giant component and phase transition; maximum degree; clique number. Other aspects of random graphs such as: connectivity, chromatic number, bipartite matchings, sharp thresholds. Percolation Theory: nontriviality of the phase transition; lattice animals, percolation on trees, uniqueness of the infinite component, properties and interpretation of the percolation probability. Other aspects of percolation theory such as: the critical value on the square lattice, selfavoiding walk, random resistor networks. Further topics in discrete probability may be considered such as: invariant distributions, bounds and cutoff phenomena for Markov chain mixing times and examples thereof; entropy, noiseless coding, discrete memoryless channel in information theory. 
Programme availability: 
MA40239 is Optional on the following programmes:Department of Mathematical Sciences
