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## MA40239: Discrete probability

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: non-triviality 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, self-avoiding walk, random resistor networks.
Further topics in discrete probability may be considered such as: invariant distributions, bounds and cut-off 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
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 3
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 3
• USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 3
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 3
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 4
• USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• TSMA-AFM09 : MSc Mathematical Sciences (Full-time) - Year 1
• TSMA-AFM08 : MSc Modern Applications of Mathematics (Full-time) - Year 1
• TSMA-AFL02 : PG Dip Modern Applications of Mathematics (Full-time) - Year 1

Notes:
* This unit catalogue is applicable for the 2012/13 academic year only. Students continuing their studies into 2013/14 and beyond should not assume that this unit will be available in future years in the format displayed here for 2012/13.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.