  We provide a sufficient condition on the isoperimetric properties of a
regular graph $G$ of growing degree $d$, under which the random subgraph $G_p$
typically undergoes a phase transition around $p=\frac{1}{d}$ which resembles
the emergence of a giant component in the binomial random graph model $G(n,p)$.
We further show that this condition is tight.
  More precisely, let $d=\omega(1)$, let $\epsilon>0$ be a small enough
constant, and let $p \cdot d=1+\epsilon$. We show that if $C$ is sufficiently
large and $G$ is a $d$-regular $n$-vertex graph where every subset $S\subseteq
V(G)$ of order at most $\frac{n}{2}$ has edge-boundary of size at least $C|S|$,
then $G_p$ typically has a unique linear sized component, whose order is
asymptotically $y(\epsilon)n$, where $y(\epsilon)$ is the survival probability
of a Galton-Watson tree with offspring distribution Po$(1+\epsilon)$. We
further give examples to show that this result is tight both in terms of its
dependence on $C$, and with respect to the order of the second-largest
component.
  We also consider a more general setting, where we only control the expansion
of sets up to size $k$. In this case, we show that if $G$ is such that every
subset $S\subseteq V(G)$ of order at most $k$ has edge-boundary of size at
least $d|S|$ and $p$ is such that $p\cdot d \geq 1 + \epsilon$, then $G_p$
typically contains a component of order $\Omega(k)$.
