  A pair of flat parallel surfaces, each freely diffusing along the direction
of their separation, will eventually come into contact. If the shapes of these
surfaces also fluctuate, then contact will occur when their centers of mass
remain separated by a nonzero distance $\ell$. Here we examine the statistics
of $\ell$ at the time of first contact for surfaces that evolve in time
according to the Edwards-Wilkinson equation. We present a general approach to
calculate its probability distribution and determine how its most likely value
$\ell^*$ depends on the surfaces' lateral size $L$. We are motivated by an
interest in the motion of interfaces between two phases at conditions of
thermodynamic coexistence, and in particular the annihilation of domain wall
pairs under periodic boundary conditions. Computer simulations of this scenario
verify the predicted scaling behavior in two and three dimensions. In the
latter case, slow growth where $\ell^\ast$ is an algebraic function of $\log L$
implies that slab-shaped domains remain topologically intact until $\ell$
becomes very small, contradicting expectations from equilibrium thermodynamics.
