  We study the capacity region of two-user erasure interference channels with
local delayed channel state information at the transmitters. In our model,
transmitters have local mismatched outdated knowledge of the channel gains. We
propose a transmission strategy that only relies on the delayed knowledge of
the {\it outgoing} links at each transmitter and achieves the outer-bound for
the scenario in which transmitters learn the entire channel state with delay.
Our result reveals the subset of the channel state information that affects the
capacity region the most.
  We also identify cases in which local delayed knowledge of the channel state
does not provide any gain over the zero knowledge assumption. To do so, we
revisit a long-known intuition about interference channels that as long as the
marginal distributions at the receivers are conserved, the capacity remains the
same. We take this intuition and impose a certain spatial correlation among
channel gains such that the marginal distributions remain unchanged. Then we
provide an outer-bound on the capacity region of the channel with correlation
that matches the capacity region when transmitters do not have access to
channel state information.
