Strong approximation for the supermarket model

We prove three strong approximation theorems for the .supermarket. or .join the shortest queue. model.a law of large numbers, a jump process approximation and a central limit theorem. The estimates are carried through rather explicitly, and rely in part on couplings. This allows us to approximate each of the infinitely many components of the process in its own scale and to exhibit a cut-off in the set of active components which grows slowly with the number of servers.