Tejas Iyer

Percolation

TU Berlin, Winter 2025/26. Lecturer: Tejas Iyer. 5 ECTS, taught in English.

Materials

Organisation

Lectures were held on Tuesdays 16:00–18:00 in MA 548, from 14 October 2025 to 10 February 2026. Assessment was by oral exam.

Description

Percolation models are probabilistic models of random graphs where nodes (often called sites) or links (called bonds) are retained with 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 or information on networks, the connectivity of networks, and the propagation of fluids or forest fires. Applied students will gain intuition about critical thresholds, cluster formation and the stochastic behaviour underlying complex systems.

Syllabus

The course covers the classical model of Bernoulli bond percolation. Core topics:

  • Existence of the phase transition: the Peierls argument.
  • Probabilistic tools: increasing coupling, the Harris–FKG inequality, the van den Berg–Kesten and Reimer inequalities, the Margulis–Russo formula.
  • The infinite cluster: the Burton–Keane argument.
  • Exponential decay in the subcritical regime.
  • Kesten’s theorem.

Flexible and advanced topics, depending on progress and interest:

  • Conformal invariance of two-dimensional percolation.
  • Critical behaviour.
  • Percolation on trees (models of hierarchical networks).
  • First-passage percolation (modelling time-to-spread phenomena).
  • The Ising model.

Prerequisites

Probability I. Probability II is highly encouraged, or may be taken alongside this course. Please email me if you have any doubts.

Learning outcomes

Beyond preparing students for research, the course develops advanced probabilistic reasoning and problem-solving skills applicable in tech, finance and other data-driven fields, and the ability to communicate complex ideas. Students will be able to:

  • Explain and prove key results in percolation theory, such as the existence of the phase transition, uniqueness of the infinite cluster and exponential decay.
  • Apply probabilistic tools such as correlation inequalities (Harris–FKG) and coupling arguments.
  • Reason about new problems in percolation and related models.

Assessment

A 45-minute oral exam, which examines your ability to:

  1. Explain and prove key results in a one-on-one interview setting (approximately three quarters of the exam).
  2. Solve new problems using the tools developed in the course, by reasoning about problems not seen before (approximately one quarter of the exam).

Theses

Students may use this course as preparation for a Bachelor’s or Master’s thesis with me. Wolfgang König is also happy to supervise Bachelor’s and Master’s theses on this topic.

Key references

  • G. Grimmett, Percolation, 2nd ed. Springer, 1999.
  • B. Bollobás and O. Riordan, Percolation. Cambridge University Press, 2006.