Lecturer: Tejas Iyer
Class format/ location: Lectures every Tuesday 16-18 in MA548, starting 14/10/2025 until 10/02/2026. (Except two weeks around Christmas - see also Moses and ISIS)
Moses: Course entry
ISIS: Course entry
Contact: After lectures, or email me - firstname.lastname@wias-berlin.de.
Language: English
Credits: 5 ECTS
Course description
Percolation models are probabilistic models of random graphs where nodes (often called sites) or links (called bonds) are retained according to a certain probability. Originally introduced to model fluid flow in porous media, percolation has become the canonical example of a probabilistic phase transition: below a critical parameter, all connected clusters are finite, while above it, a single infinite cluster emerges. Its mathematical simplicity makes it a foundation for more complex models in statistical physics and probability.
In applications, percolation provides conceptual insights and probabilistic tools for understanding tipping points in the spread of epidemics (disease or information) on networks, connectivity of networks, or the propagation of fluid or forest fires. Applied students will gain intuition about critical thresholds, cluster formation, and stochastic behaviour underlying complex systems.
Syllabus
This course will delve into topics concerning the classical model of Bernoulli bond percolation.
Core Topics:
Existence of the phase transition: Peierls argument.
Usage of probabilistic tools: increasing coupling, Harris-FKG inequality, Van den Berg-Kesten and Reimer inequalities, Margulis-Russo formula.
Infinite cluster: Burton-Keane argument.
Exponential decay in the subcritical regime
Kesten's theorem.
Flexible/ advanced topics (depending on class progress/interest):
Conformal invariance of two-dimensional percolation
Critical behaviour
Percolation on trees (models of heirarchical networks)
First passage percolation (modelling time-to-spread phenomena)
Ising model
Prerequisites:
Probability I.
Probability II is highly encouraged or may be taken concurrently with this lecture.
Please contact me by email if you have any doubts.
Learning outcomes
Beyond preparing students for research, this course will develop skills in advanced probabilistic reasoning and problem solving skills applicable in tech, finance and other data-driven fields. Students will also develop important skills in communication of complex ideas.
More precisely, on satisfying the requirements of this course, students will be well-equipped to:
Explain and prove key results in percolation theory (e.g. existence of the phase transition, uniqueness of the infinite cluster, exponential decay)
Apply probabilistic tools such as correlation inequalities (BK, FKG) and coupling arguments.
Reason about new problems in percolation and related models.
Assessment: The assessment for this course consists of a 45 minute oral exam. This will examine your abilities to:
Explain and prove key results in a one-on-one interview setting (approximately 3/4 of the exam).
Solve new problems using the tools developed in the course, by reasoning in the interview setting about problem(s) not seen before (about 1/4 of the exam).
Students may use this course in preparation for a Bachelor's or Master's thesis with me. Wolfgang König is happy to supervise bachelors and masters theses in this topic until 2032.
Resources
Lecture notes will be provided as the course progresses, along with links to external resources.
Key references (including most of what we will cover):
Grimmett, G. (1999). Percolation (2nd ed.). Springer.
Bollobás, B., & Riordan, O. (2006). Percolation. Cambridge University Press.