Mathematics – Probability
Scientific paper
2006-10-05
Annals of Applied Probability 2006, Vol. 16, No. 3, 1297-1318
Mathematics
Probability
Published at http://dx.doi.org/10.1214/105051606000000330 in the Annals of Applied Probability (http://www.imstat.org/aap/) by
Scientific paper
10.1214/105051606000000330
We present an improved coupling technique for analyzing the mixing time of Markov chains. Using our technique, we simplify and extend previous results for sampling colorings and independent sets. Our approach uses properties of the stationary distribution to avoid worst-case configurations which arise in the traditional approach. As an application, we show that for $k/\Delta >1.764$, the Glauber dynamics on $k$-colorings of a graph on $n$ vertices with maximum degree $\Delta$ converges in $O(n\log n)$ steps, assuming $\Delta =\Omega(\log n)$ and that the graph is triangle-free. Previously, girth $\ge 5$ was needed. As a second application, we give a polynomial-time algorithm for sampling weighted independent sets from the Gibbs distribution of the hard-core lattice gas model at fugacity $\lambda <(1-\epsilon)e/\Delta$, on a regular graph $G$ on $n$ vertices of degree $\Delta =\Omega(\log n)$ and girth $\ge 6$. The best known algorithm for general graphs currently assumes $\lambda <2/(\Delta -2)$.
Hayes Thomas P.
Vigoda Eric
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