describe the action of the dilatation operator on restricted Schur polynomials. The main result of this section is formula (\[final\]) which gives a very explicit description of the action of the dilatation operator. In section 3 we will describe, in broad terms, the set of operators we consider and explain how the dynamics of the large $N$ limit simplifies. In section 4 we explain how to construct the projectors needed to evaluate the action of the dilatation operator. A very detailed description of the specific operators we consider is given in section 5; we also describe a limit in which our system of two giants should be well described as a system of $D3$ brane giants plus open strings. In this limit we see that the dilatation operator reduces to a lattice version of the second derivative, with the Young diagram labels of the restricted Schur polynomials defining the lattice. In section 6 we present our numerical results and draw some general conclusions from them in section 7. The explicit result for the dilatation operator is given in Appendix A; the intertwiners which enter into the expression for the dilatation operator are described in Appendix B.

Action of the Dilatation 