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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

Concentrated Advanced Course on

Percolation Theory and Applications in Statistical Mechanics

Lectures by

Olle Häggström (Chalmers, Gothenburg)


Johan Jonasson (Chalmers, Gothenburg)

Monday August 14 - Friday August 18, 2000

University of Copenhagen

In the above-mentioned week, MaPhySto organized a Concentrated Advanced Course on Percolation Theory and Applications in Statistical Mechanics. The course took place at the Department of Mathematical Sciences, University of Copenhagen. The course was organized by Martin Jacobsen (Copenhagen).

Content of course

The goal of this course was to present the basics of percolation theory, plus some applications in statistical mechanics and other recent developments. No particular prerequisites are needed, other than familiarity with probability founded on measure theory.

The material can roughly be divided into three parts:

I. The standard percolation model

Site and bond percolation on the integer lattice Zd. The threshold phenomenon. Characteristic features of subcritical, critical, and supercritical percolation, including uniqueness of the infinite cluster in the supercritical case. The exact critical value for bond percolation in two dimensions.

II. Applications in statistical mechanics

Gibbs systems, such as Ising, Potts and Widom-Rowlinson models. Phase transitions in these models. Percolation techniques, including random-cluster representations and disagreement percolation, for the analysis of phase transitions.

III. Percolation on other graph structures

An overview of the recent substantial developments in the theory of percolation on graph structures other than the Zd, such as trees and Cayley graphs. The importance of amenability and unimodularity. Newly discovered phenomena in the nonamenable setting, such as the double threshold phenomenon, and properties of infinite clusters in the intermediate regime.

For part I, any of Grimmett's standard references

G.R. Grimmett
Percolation, Springer, 1989; or 2nd edition from 1999.
G.R. Grimmett
Percolation and disordered systems, in Lectures on probability theory and statistics (Saint-Flour, 1996), pp 153--300, Springer 1997.

will be fine.

Part II will use

H.-O. Georgii, O. Häggström and C. Maes
The random geometry of equilibrium phases, in Phase Transitions and Critical Phenomena (C. Domb and J.L. Lebowitz, eds), Academic Press, to appear.

The material for part III is more scattered, but some papers that are likely to be used are

O. Häggström, Y. Peres and R. Schonmann
Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness, in Perplexing Probability Problems: Festschrift in Honor of Harry Kesten (M. Bramson and R. Durrett, eds), pp 69-90, Birkhäuser, 1999.
J. Jonasson
The random cluster model on a general graph and a phase transition characterization of nonamenability, Stoch. Proc. Appl. 79 (1999), 335-354.
I. Benjamini, R. Lyons, Y. Peres and O. Schramm
Critical percolation on any nonamenable group has no infinite clusters, Ann. Probab., to appear.

Provisional Schedule

The course started on Monday morning, with registration from 9.00 to 10.00. During the lunch break there were ample time for discussions.

TimeLecturerLecture title
Monday 10.15-12.00 OH Introduction to percolation on Zd
Monday 14.15-16.00 JJ Exact critical values in two dimensions
Tuesday 10.15-12.00 JJ Ising and Potts models. Gibbs measures. Phase transitions via random-cluster representations.
Tuesday 14.15-16.00 OH Other connections between percolation and Gibbs measures
Wednesday 10.15-12.00 JJ Continuum percolation
Wednesday 14.15-16.00 OH The Widom-Rowlinson model
Thursday 10.15-12.00 JJ Percolation on trees
Thursday 14.15-16.00 OH Percolation on nonamenable graphs
Friday 10.15-12.00 JJ The Ising model on nonamenable graphs
Friday 14.15-16.00 OH Overview of other topics in percolation theory


More Information

Do not hesitate to contact the MaPhySto secretariat (at maphysto@maphysto.dk) or the local organizer Martin Jacobsen for more information.

This document, http://www.maphysto.dk/oldpages/events/PTandSM2000/index.html, was last modified January 19, 2004. Questions or comments to the contents of this document should be directed to maphysto@maphysto.dk.