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:
- Percolation on graphs. Definitions and basic properties
- Harris' inequality
- van den Berg-Kesten inequality, Reimer's inequality
- Russo's formula
- Burton-Keane Theorem (Aizenman-Kesten-Newman)
- Exponential decay of correlations in sub-critical regime
- Planar percolation: Russo-Seymour-Welsh theory
- Planar percolation: the Harris-Kesten theorem
- Conformal invariance: Cardy-Smirnov formula on the triangular lattice
- Percolation on groups
- Critical percolation on non-amenable groups: BLPS
- Percolation on finite graphs: The Erdos-Renyi model
B. Bollobas and O. Riordan.
- G. Grimmett.
- R. Lyons with Y. Peres.
Probability on Trees and Networks.
- Hugo Duminil-Copin's
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(More lectures will be added.)