  It has long been known that random regular graphs are with high probability
good expanders. This was first established in the 1980s by Bollob\'as by
directly calculating the probability that a set of vertices has small expansion
and then applying the union bound.
  In this paper we improve on this analysis by relying on a simple high-level
observation: if a graph contains a set of vertices with small expansion then it
must also contain such a set of vertices that is locally optimal, that is, a
set whose expansion cannot be made smaller by exchanging a vertex from the set
with one from the set's complement. We show that the probability that a set of
vertices satisfies this additional property is significantly smaller. Thus,
after again applying the union bound, we obtain improved lower bounds on the
expansion of random $\Delta$-regular graphs for $\Delta\ge 4$. In fact, the
gains from this analysis increase as $\Delta$ grows, a fact we explain by
extending our technique to general $\Delta$. Thus, in the end we obtain an
improvement not only for some small special cases but on the general asymptotic
bound on the expansion of $\Delta$-regular graphs given by Bollob\'as.
