  The problem of completing a large low rank matrix using a subset of revealed
entries has received much attention in the last ten years. The main result of
this paper gives a necessary and sufficient condition, stated in the language
of graph limit theory, for a sequence of matrix completion problems with
arbitrary missing patterns to be asymptotically solvable. It is then shown that
a small modification of the Cand\`es-Recht nuclear norm minimization algorithm
provides the required asymptotic solution whenever the sequence of problems is
asymptotically solvable. The theory is fully deterministic, with no assumption
of randomness. A number of open questions are listed.
