  Suppose that we are given an arbitrary graph $G=(V, E)$ and know that each
edge in $E$ is going to be realized independently with some probability $p$.
The goal in the stochastic matching problem is to pick a sparse subgraph $Q$ of
$G$ such that the realized edges in $Q$, in expectation, include a matching
that is approximately as large as the maximum matching among the realized edges
of $G$. The maximum degree of $Q$ can depend on $p$, but not on the size of
$G$.
  This problem has been subject to extensive studies over the years and the
approximation factor has been improved from $0.5$ to $0.5001$ to $0.6568$ and
eventually to $2/3$. In this work, we analyze a natural sampling-based
algorithm and show that it can obtain all the way up to $(1-\epsilon)$
approximation, for any constant $\epsilon > 0$.
  A key and of possible independent interest component of our analysis is an
algorithm that constructs a matching on a stochastic graph, which among some
other important properties, guarantees that each vertex is matched
independently from the vertices that are sufficiently far. This allows us to
bypass a previously known barrier towards achieving $(1-\epsilon)$
approximation based on existence of dense Ruzsa-Szemer\'edi graphs.
