---------- Forwarded message ----------
From: Walter Schachermayer <wschach(a)fam.tuwien.ac.at>
---------- Forwarded message ----------
Date: Fri, 8 Oct 2004 13:40:19 -0700
From: Edwin Perkins <perkins(a)math.ubc.ca>
To: wschach(a)fam.tuwien.ac.at
Subject: 2005 Summer School in Probability June 6-30, U. British Columbia
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.
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(a)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 at the above address.
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(a)math.ubc.ca.
Sincerely, David Brydges and Ed Perkins.
For summer 2005 the courses will be given by Yuval Peres and Gordon Slade,
and will run from 6 June 2005 - 30 June 2005
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Yuval Peres (Math 610D)
Title: Mixing for Markov Chains and Spin Systems
Instructor: Yuval Peres, UC Berkeley
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:
* Connections between mixing in time and space for spin systems
* Comparison of updates at random locations and systematic scans
* The cutoff phenomenon for distance to stationarity
Gordon Slade (Math 609D)
``The lace expansion and its applications"
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.
Lecture notes for the course are available at
http://www.math.ubc.ca/~slade/sf_v1.ps.gz