  Consider a finite irreducible Markov chain with invariant distribution $\pi$.
We use the inner product induced by $\pi$ and the associated heat operator to
simplify and generalize some results related to graph partitioning and the
small-set expansion problem. For example, Steurer showed a tight connection
between the number of small eigenvalues of a graph's Laplacian and the
expansion of small sets in that graph. We give a simplified proof which
generalizes to the nonregular, directed case. This result implies an
approximation algorithm for an "analytic" version of the Small-Set Expansion
Problem, which, in turn, immediately gives an approximation algorithm for
Small-Set Expansion. We also give a simpler proof of a lower bound on the
probability that a random walk stays within a set; this result was used in some
recent works on finding small sparse cuts.
