interesting examples including LLM geometries [@Lin:2004nb; @Balasubramanian:2005mg] and the near horizon geometry of a bound state of giant gravitons[@shahin]. [*The results are remarkably simple*]{}. Indeed, as an example, for ${1\over 2}$ BPS-correlators in the presence of $M$ giant gravitons with $M$ of order $N$, [@Koch:2008ah; @Koch:2008cm; @Koch:2009jc] showed that the usual ${1\over N}$ expansion is replaced by a ${1\over M+N}$ expansion. Further, if one expands the exact correlators (which because they are ${1\over 2}$ BPS do not depend on $g_{YM}^2$ but only on $N$ or $N+M$) the expansion coefficients for correlators in the background of $M$ giants are exactly the same as the expansion coefficients for correlators with no giants present! This remarkably simple result was confirmed holographically[@Maldacena:1997re] by matching to graviton dynamics in the LLM geometries using the formalism of [@Skenderis:2007yb]. For near-BPS operators corresponding to BMN loops[@Berenstein:2002jq] it was argued in [@Chen:2007gh; @us; @hai1] that the usual ’t Hooft coupling $g_{YM}^2N$ is replaced by the effective ’t Hooft coupling $g_{YM}^2 (N+M)$. For additional interesting related studies see[@shahin; @hai2].

In this article we will consider the problem of computing the anomalous dimension of an operator with a bare dimension of order $N$. To answer this question, we need to 