In this paper we examine the importance of the choice of metric in path coupling, and the relationship of this to \emph{stopping time analysis}. We give strong evidence that stopping time analysis is no more powerful than standard path coupling. In particular, we prove a stronger theorem for path coupling ... more >>>

We give a dichotomy theorem for the problem of counting homomorphisms to
directed acyclic graphs. $H$ is a fixed directed acyclic graph.
The problem is, given an input digraph $G$, how many homomorphisms are there
from $G$ to $H$. We give a graph-theoretic classification, showing that
for some digraphs $H$, ... more >>>

We consider Glauber dynamics on finite spin systems.
The mixing time of Glauber dynamics can be bounded
in terms of the influences of sites on each other.
We consider three parameters bounding these influences ---
$\alpha$, the total influence on a site, as studied by Dobrushin;
$\alpha'$, the total influence ... more >>>

We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree $\Delta$ of ... more >>>

We study a simple Markov chain, known as the Glauber dynamics, for generating a random <i>k</i>-coloring of a <i>n</i>-vertex graph with maximum degree &Delta;. We prove that the dynamics converges to a random coloring after <i>O</i>(<i>n</i> log <i>n</i>) steps assuming <i>k</i> &ge; <i>k</i>₀ for some absolute constant <i>k</i>₀, and either: ... more >>>