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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, February 22, 2000}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 p.m. in 1st Floor Lounge)}

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\centerline{\sl David Aldous}
\centerline{\sl Department of Statistics}
\centerline{\sl UC Berkeley}
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\centerline{\bf The East ($=$ one-dimensional asymmetric constrained
Ising) model: towards rigorous results}
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This is joint work in progress with Persi Diaconis, inspired by Hans C.
Andersen (Chemistry).  The model
is an interacting particle system with Bernoulli($p$) stationary
distribution, in which a site can only change
state (occupied to unoccupied or vice versa) when the site immediately to
its left is occupied.  When the density parameter $p$ is small the process
takes a long time to move far away from its initial configuration.  It has
been conjectured that the relaxation time ( = 1/spectral gap) is order
$(1/p)^{\log 1/p}$ as $p$ goes to 0. The model is methodologically interesting
because it seems intermediate between the typical Ising-type models of
statistical physics and the more combinatorial finite Markov chain models
whose relaxation times have been studied by Diaconis et al.  Our proofs of
rigorous bounds combine several ingredients:

(i) the Poincare comparison method, whose use involves analyzing the
combinatorial structure of transition-paths

(ii) invention of a "long-range wave" process for comparison purposes, and
its analysis via coupling and exponential martingales 

(iii) for the lower bound, invention of an approximate "dual" process of
coalescing random jumps and its analysis.

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