  It is known that many different types of finite random subgraph models
undergo quantitatively similar phase transitions around their percolation
thresholds, and the proofs of these results rely on isoperimetric properties of
the underlying host graph. Recently, the authors showed that such a phase
transition occurs in a large class of regular high-dimensional product graphs,
generalising a classic result for the hypercube.
  In this paper we give new isoperimetric inequalities for such regular
high-dimensional product graphs, which generalise the well-known isoperimetric
inequality of Harper for the hypercube, and are asymptotically sharp for a wide
range of set sizes. We then use these isoperimetric properties to investigate
the structure of the giant component $L_1$ in supercritical percolation on
these product graphs, that is, when $p=\frac{1+\epsilon}{d}$, where $d$ is the
degree of the product graph and $\epsilon>0$ is a small enough constant.
  We show that typically $L_1$ has edge-expansion $\Omega\left(\frac{1}{d\ln
d}\right)$. Furthermore, we show that $L_1$ likely contains a linear-sized
subgraph with vertex-expansion $\Omega\left(\frac{1}{d\ln d}\right)$. These
results are best possible up to the logarithmic factor in $d$.
  Using these likely expansion properties, we determine, up to small
polylogarithmic factors in $d$, the likely diameter of $L_1$ as well as the
typical mixing time of a lazy random walk on $L_1$. Furthermore, we show the
likely existence of a path of length $\Omega\left(\frac{n}{d\ln d}\right)$.
These results not only generalise, but also improve substantially upon the
known bounds in the case of the hypercube, where in particular the likely
diameter and typical mixing time of $L_1$ were previously only known to be
polynomial in $d$.
