  It has been recently proposed that string theory in the background of a plane
wave corresponds to a certain subsector of the N=4 supersymmetric Yang-Mills
theory. This correspondence follows as a limit of the AdS/CFT duality. As a
particular case of the AdS/CFT correspondence, it is a priori a strong/weak
coupling duality. However, the predictions for the anomalous dimensions which
follow from this particular limit are analytic functions of the 't Hooft
coupling constant $\lambda$ and have a well defined expansion in the weak
coupling regime. This allows one to conjecture that the correspondence between
the strings on the plane wave background and the Yang-Mills theory works at the
level of perturbative expansions.
  In our paper we perform perturbative computations in the Yang-Mills theory
that confirm this conjecture. We calculate the anomalous dimension of the
operator corresponding to the elementary string excitation. We verify at the
two loop level that the anomalous dimension has a finite limit when the R
charge $J\to \infty$ keeping $\lambda/J^2$ finite. We conjecture that this is
true at higher orders of perturbation theory. We show, by summing an infinite
subset of Feynman diagrams, under the above assumption, that the anomalous
dimensions arising from the Yang-Mills perturbation theory are in agreement
with the anomalous dimensions following from the string worldsheet sigma-model.
