Ben Gurion University
Department of Mathematics




Course: 201.2.0101           Spring 2013



The course is intended for anyone with a basic course in probability.

This course concerns the physical notion of phase transition, specifically in the model known as "percolation". We will review the main mathematical results regarding percolation and its related counterparts the Ising model, Potts model and Fortuin-Kasteleyn cluster model, starting from works of Ising and Peierls in the beginning of the 20th century and culminating in modern work of Smirnov.
The topics in the course are, time permitting:

  1. Percolation on graphs. Definitions and basic properties
  2. Harris' inequality
  3. van den Berg-Kesten inequality, Reimer's inequality
  4. Russo's formula
  5. Burton-Keane Theorem (Aizenman-Kesten-Newman)
  6. Exponential decay of correlations in sub-critical regime
  7. Planar percolation: Russo-Seymour-Welsh theory
  8. Planar percolation: the Harris-Kesten theorem
  9. Conformal invariance: Cardy-Smirnov formula on the triangular lattice
  10. Percolation on groups
  11. Critical percolation on non-amenable groups: BLPS
  12. Percolation on finite graphs: The Erdos-Renyi model

Recommended Literature:

  1. B. Bollobas and O. Riordan. Percolation.
  2. G. Grimmett. Percolation.
  3. R. Lyons with Y. Peres. Probability on Trees and Networks.
  4. Hugo Duminil-Copin's Lecture Notes (in French).



Lecture Notes

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