155 research outputs found
Making Markov chains less lazy
The mixing time of an ergodic, reversible Markov chain can be bounded in
terms of the eigenvalues of the chain: specifically, the second-largest
eigenvalue and the smallest eigenvalue. It has become standard to focus only on
the second-largest eigenvalue, by making the Markov chain "lazy". (A lazy chain
does nothing at each step with probability at least 1/2, and has only
nonnegative eigenvalues.)
An alternative approach to bounding the smallest eigenvalue was given by
Diaconis and Stroock and Diaconis and Saloff-Coste. We give examples to show
that using this approach it can be quite easy to obtain a bound on the smallest
eigenvalue of a combinatorial Markov chain which is several orders of magnitude
below the best-known bound on the second-largest eigenvalue.Comment: 8 page
Rapid Mixing for Lattice Colorings with Fewer Colors
We provide an optimally mixing Markov chain for 6-colorings of the square
lattice on rectangular regions with free, fixed, or toroidal boundary
conditions. This implies that the uniform distribution on the set of such
colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet
has a finite correlation length and a unique Gibbs measure at zero temperature.
Four and five are now the only remaining values of q for which it is not known
whether there exists a rapidly mixing Markov chain for q-colorings of the
square lattice.Comment: Appeared in Proc. LATIN 2004, to appear in JSTA
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