  Canonical paths is one of the most powerful tools available to show that a
Markov chain is rapidly mixing, thereby enabling approximate sampling from
complex high dimensional distributions. Two success stories for the canonical
paths method are chains for drawing matchings in a graph, and a chain for a
version of the Ising model called the subgraphs world. In this paper, it is
shown that a subgraphs world draw can be obtained by taking a draw from
matchings on a graph that is linear in the size of the original graph. This
provides a partial answer to why canonical paths works so well for both
problems, as well as providing a new source of algorithms for the Ising model.
For instance, this new reduction immediately yields a fully polynomial time
approximation scheme for the Ising model on a bounded degree graph when the
magnitization is bounded away from 0.
