Prof. Dr. Volker Schmidt
Dipl-Math. Christian Hirsch
Time and place
Thursday, 14-16 (HeHo 18, Room 220)
Tuesday, 12:30-14:00 (HeHo 18, Room 220)
The intended language of the lecture will be English. However if students unanimously express the wish that the lecture be held in German, this will also be possible.
2 hours lecture + 2 hours exercises
Credit points: 6
Master students in "Mathematik", "Wirtschaftsmathematik" and "Mathematische Biometrie"; Master students in Finance
Bachelor students in "Mathematik", "Wirtschaftsmathematik" and "Mathematische Biometrie" are also encouraged to participate: An accreditation as supplementary module ("Zusatzmodul") is possible
Stochastic networks and, in particular, random geometric graphs have been studied since more than 50 years. This part of stochastic geometry continues to produce intriguing questions that prove to be a fruitful topic of ongoing mathematical research. Another reason for the success of these networks is due to their importance in applications - at the macroscopic as well as the microscopic scale. Road systems, telecommunication and electricity networks, pore space and solid phase of advanced energy materials (used e.g. in Li-ion batteries, fuel/solar cells), as well as biological structures (such as subcellular networks) can be modelled by appropriate types of geometric graphs.
Participants of this lecture will learn the basics of percolation theory both in the case of discrete as well as continuum state space. After being familiar with the important notion of phase transition, classical topics such as relationships between bond and site percolation, uniqueness of the infinite cluster or cluster size behaviour in the subcritical case will be treated and basic results on dependent percolation will be considered. A solid understanding of these topics allows us to proceed to the more complex case of continuum percolation, where many analogues of the previously encountered results can be derived.
Later on we can proceed to more advanced topics such as Erdős–Rényi graphs, oriented percolation, first passage percolation, minimal spanning forests or (random) Euclidean optimization problems.
Requirements and Exam
Active participation in the exercise classes is a prerequisite for admission to the final exam. All students who want to participate in the exam and who have participated actively in the exercise classes are kindly asked to register for the 'Vorleistung' in the LSF-'Hochschulportal'.
The final exam will be held on Saturday, February 25, 2012.
The exam takes place 9:00-11:00 in room H8 (main university building). Please be there 10 minutes in advance so that we can start on time. Participants may bring one handwritten A4 sheet (both sides) containing useful results from the lecture and exercise classes. Copies and printouts are not allowed.
We provide preliminary notes (in German) that can be used to get an overview over the most important topics considered in the lecture. These notes will be updated as the course proceeds. Please be aware that these notes only cover parts of the lecture and that the additional information provided in the lecture will be relevant for the final exam.
- Bollobás, Riordan: Percolation. Cambridge University Press, 2006
- Grimmett: Percolation. New York: Springer, 1989
- Grimmett: Probability on Graphs - Random Processes on Graphs and Lattices. Cambridge University Press, 2010
- Hofstad, van der: Random graphs and complex networks
- Lyons, Peres: Probability on trees and networks. Cambridge University Press, 2011
- Penrose: Random Geometric Graphs. Oxford University Press, 2007.
- Steele: Probability theory and combinatorial optimization. SIAM, Philadelphia (1997).
- Yukich: Limit Theorems in Discrete Stochastic Geometry
- Yukich: Probability theory of classical Euclidean optimization problems. LNM 1675, Springer (1998).