PIMS-MITACS Summer School in Probability June 6 - June 30, 2005 PIMS-UBC

As part of our PIMS Collaborative Research Group in Probability and Statistical Physics we will again be running two advanced graduate courses at UBC in the summer of 2005. The lectures in 2005 will take place from June 6 to June 30 and be given by Yuval Peres, U Cal. Berkeley, and Gordon Slade, UBC. The course descriptions are below. We plan these to be official courses at UBC and so graduate students at universities in W. Canada can receive credit for them through the Western Deans Agreement. There will be total of 30 hours of lectures in each course. Support for these courses comes from the Pacific Institute for the Mathematical Sciences and the Department of Mathematics at UBC.

Those interested in attending these courses (graduate students, pdf's , faculty members) are encouraged to sign up asap at our website http://www.pims.math.ca/science/2005/ssprob/ as there will be limited space in the lecture rooms. Registration at this site should be available by Oct. 8

There will be some financial support available for a limited number of graduate students and postdoctoral fellows who would like to attend. This will basically cover the cost of a dorm room for the duration of the course. Applications for support should consist of a brief letter of application, a cv of the prospective student/pdf and a letter from the applicant's supervisor all of which may be emailed to ssprob@pims.math.ca. Plain text is preferred. The deadline for applications for financial support is Dec. 31. It would help us greatly with planning if you could let us know of your interest before Oct. 31.

If you have queries about the courses please check our website at http://www.pims.math.ca/science/2005/ssprob/ or send questions to Gordon Slade at slade@math.ubc.ca.

Sincerely, David Brydges and Ed Perkins.

Courses

Title of course: The lace expansion and its applications

Instructor: Gordon Slade (University of British Columbia)

Abstract: Several superficially simple mathematical models, such as the self-avoiding walk and percolation, are paradigms for the study of critical phenomena in statistical mechanics. It remains a major challenge for mathematical physics and probability theory to obtain a mathematically rigorous understanding of the scaling theory of these models at criticality. The lace expansion has become a powerful tool for the analysis of the critical scaling of a number of models above their upper critical dimensions, including the self-avoiding walk, lattice trees, lattice animals, oriented and non-oriented percolation, and the contact process. Results include proof of existence of critical exponents, with mean-field values, and construction of the scaling limit. For lattice trees and critical percolation, the scaling limit is described in terms of super-Brownian motion. The lectures will provide an introduction to the lace expansion and several of its applications. No previous exposure to the lace expansion will be assumed, and necessary background will be provided.

The latest version of the course lecture notes are available at http://www.math.ubc.ca/~slade.

Title of course: Mixing for Markov Chains and Spin Systems

Instructor: Yuval Peres (University of California, Berkeley)

Abstract: Given an aperiodic irreducible Markov chain on a finite state space, the rate at which it approaches its stationary distribution is intensively studied by mathematical physicists, computer scientists and probabilists. The key insight is that as we consider running the chain for longer times, we should also be considering chains on larger spaces. Two celebrated families of chains that still pose mysteries are random walks on the symmetric group (card shuffles) and "Glauber dynamics" of spin systems; canonical examples for the latter are the Ising model and graph colorings.

Planned topics:

• Markov chains and electrical networks: a brief primer.
• Probabilistic methods: coupling and strong uniform times.
• Mixing via spectral gap and isoperimetric inequalities.
• Expanders via random constructions and zigzag products.
• The Ising model and the random cluster model.
• The Ising model on trees, and its interpretations in mathematical genetics and noisy computation.
• Glauber dynamics for spin systems.
• Correlation inequalities and their implications for mixing.
• Cover times and lamplighter groups.
• Exact sampling via coupling from the past.

Three areas, teeming with unsolved problems, that we will explore:

1. Connections between mixing in time and space for spin systems
2. Comparison of updates at random locations and systematic scans
3. The cutoff phenomenon for distance to stationarity

Schedule

The first lectures on Monday, June 6 will be in the Mathematics building (Math 104) located in grid C3 on the campus map.

Slade - The lace expansion and its applications

Peres - Mixing for Markov Chains and Spin Systems

Week 1 June 6-10:

Week 2 June 13-17:

Week 3 June 20-24:

Week 4 June 27-30:

During the summer school, two classes per week (normally on Tuesday and Thursday afternoons) will be reserved for up to three 25-minute presentations by participants in the summer school, and occasional lectures by other visitors to PIMS/UBC.