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Institute of Mathematical Finance
Prof. Dr. Alexander Lindner
Martin Drapatz
Master Seminar Percolation
The institute of Mathematical Finance organizes in the winter term 14/15 a Master Seminar
on Percolation. To attend this seminar only knowledge in probability theory is necessary.
The seminar will be hold weekly and the maximal number of participants is restricted to 15
students. Registration is possible by mail to [email protected].
Percolation theory was founded in order to model the flow of a liquid in a porous medium
with randomly blocked channels. It can be seen as the study of the structure of random
subgraphs of a specific graph. In most of the situations a lattice graph is considered where
either the vertices or the edges are independently open or closed according to a Bernoulli
distribution with same probability p ∈ (0, 1).
Figure 1: simulated bond percolation on the square lattice Z2 , p = 0.51
Typical questions one then might ask are for example: for which p ∈ (0, 1) does an open
path of infinite length (’infinite cluster’) exist with probability one? is it possibile that more
than one infinite cluster exists? Denote the open cluster containing a vertex x by Cx and
by Pp the associated probability measure where the vertices/edges are open with probability
p ∈ (0, 1). Define further
θx (p) := Pp (|Cx | = ∞).
What is the critical probability
pc = sup{p ∈ (0, 1) : θx (p) = 0} ?
Below some suggestions for topics are listed:
1. Basic concepts and results, Chapter 1 in Bollob´as and Riordan [2], pp. 1-35.
Present the basic concepts which are introduced in Chapter 1. In particular, define the
critical probabilities pbH , pcH , pbT and psT and show their relationships.
2. Probabilitic tools, Chapter 2 in Bollob´as and Riordan [2], pp. 36-49.
Introduce the probabilistic model for a random graph (’the weighted random cube’) and the
’box’-operation. Prove the Harris Lemma and the FKG inequality.
3. Harris-Kesten Theorem 1, Sections 3.0, 3.1 and 3.2 in Bollob´as and Riordan [2], pp.
50-62.
Introduce the dual graph, explain the Russo-Seymour-Welsh method and prove the Harris
Theorem.
4. Harris-Kesten Theorem 2, Sections 3.3, 3.4, 3.5 and 3.6 in Bollob´as and Riordan [2],
pp. 63-78.
Present the Kesten Theorem and show that Pp (|Cx | ≥ n) ≤ exp(−an) for some a = a(p) > 0
and p < pbH = 1/2, i.e. the probability that the cluster size exceeds n decays exponentially in
n if percolation does not occur. Show that if percolation occurs the decay is subexponential.
5. Exponential decay and critical probabilites 1, Sections 4.1 and 4.2 in Bollob´as and
Riordan [2], pp. 78-90.
Introduce oriented site percolation and present the results of these sections.
6. Exponential decay and critical probabilites 2, Section 4.3 in Bollob´as and Riordan
[2], pp. 90-104.
Prove the Menshikov Theorem.
7. Uniqueness of the infinite open cluster and critical probabilities 1, Section 5.1
and 5.2 in Bollob´as and Riordan [2], pp. 117-128.
Show uniqueness of the infinite open cluster and give an alternative proof of the Kesten
Theorem.
8. Oriented percolation 1, Liggett [7].
A Markov process is introduced and investigated to derive upper bounds for the critical
probability in oriented percolation.
9. Oriented percolation 2, Fletcher, Pearce [3] and Bishir [1].
Again the theory of Markov processes is used to derive this time lower bounds for the critical
probability in oriented percolation.
10. Subadditive ergodic theorem , Kingman [6].
The subadditive ergodic theorem is a device to prove limit theorems for processes which are
considered in first-passage Percolation. Prove the subadditive ergodic theorem.
11. First-Passage Percolation 1, Smythe and Wiermann [8] .
First-passage Percolation can be seen as generalization of percolation. Here, positive random
weights are associated to the edges and one is interested in the ’first-passage’ of a specific
vertex or a set of vertices, i.e. the path with minimal weight. Introduce the model and
present the results of Chapter 4.
12. First-Passage Percolation 2, Smythe and Wiermann [8].
With the help of the subadditive ergodic theorem limit theorems for the first-passage time
are derived. Present the results of Chapter 5.
13. First-Passage Percolation 3, Kesten [5].
Present Kesten’s results.
References
[1] J. Bishir (1963): A lower bound for the critical probability in the one-quadrant oriented
atom percolation process. Journal Royal Statistical Society 25, 401-404.
[2] B. Bollob´as and O. Riordan (2006): Percolation. Cambridge University Press, 1st
edition.
[3] F.K. Fletcher and C.E.M. Pearce (2005): Oriented site percolation, phase transitions
and probability bounds. Journal of Inequalities in Pure and Applied Mathematics 6(5).
[4] G. Grimmett (1999): Percolation. Springer, 2nd edition.
[5] Harry Kesten (1980): On the Time Constant and Path Length of First-Passage Percolation. Advances in Applied Probability 12(4), pp. 848-863.
[6] J. F. C. Kingman (1968): The Ergodic Theory of Subadditive Stochastic Processes.
Journal of the Royal Statistical Society 30(3), 499-510.
[7] T.M. Ligett (1995): Survival of Discrete Time Growth Models, with Applications to
Oriented Percolation. Ann. Appl. Probab 5(3), 613-636.
[8] R. T. Smythe and J.C. Wiermann (1978): First-Passage Percolation on the Square
Lattice. Springer, 1st edition, Berlin.

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