  We consider the Bernoulli bond percolation model in a box $\Lambda$ (not
necessarily parallel to the directions of the lattice) in the regime where the
percolation parameter is close to $1$. We condition the configuration on the
event that two opposite faces of the box are disconnected. We couple this
configuration with an unconstrained percolation configuration. The interface
edges are the edges which differ in the two configurations. We prove that,
typically, each interface edge is within a distance of order $\ln|\Lambda|$ of
another interface edge or of a pivotal edge. We derive an estimate for the law
of an edge which is far from the cut and the interface edges.
