Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree
Markov Chains in Algorithms and Statistical Physics
February 01, 2005 11:50 AM to 12:35 PM
Speakers:
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Abstract: |
Consider a low temperature stochastic Ising model in the phase
coexistence regime with Markov semigroup $P_t$. A fundamental and
still largely open problem is the understanding of the long time
behavior of $\delta_\eta P_t$ when the initial configuration $\eta$ is
sampled from a highly disordered state $\nu$ (e.g. a product Bernoulli
measure or a high temperature Gibbs measure). Exploiting recent
progresses in the analysis of the mixing time of Monte Carlo Markov
chains for discrete spin models on a regular $b$-ary tree $T^b$, we
tackle the above problem for the Ising and hard core gas (independent
sets) models on $T^b$. If $\nu$ is a biased product Bernoulli law
then, under various assumptions on the bias and on the thermodynamic
parameters, we prove $\nu$-almost sure weak convergence of
$\delta_\eta P_t$ to an extremal Gibbs measure (pure phase) and show
that the limit is approached at least as fast as a stretched
exponential of the time $t$. In the context of randomized algorithms
and if one considers the Glauber dynamics on a large, finite tree, our
results prove fast local relaxation to equilibrium on time scales much
smaller than the true mixing time, provided that the starting point of
the chain is not taken as the worst one but it is rather sampled from
a suitable distribution.
Joint work with P. Caputo |
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Lecture #10804
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