Octagons I: combinatorics and non-planar resummations

We explain how the 't Hooft expansion of correlators of half-BPS operators can be resummed in a large-charge limit in N = 4 super Yang-Mills theory. The full correlator in the limit is given by a non-trivial function of two variables: One variable is the charge of the BPS operators divided by the square root of the number N _c of colors; the other variable is the octagon that contains all the 't Hooft coupling and spacetime dependence. At each genus g in the large N _c expansion, this function is a polynomial of degree 2 g + 2 in the octagon. We find several dual matrix model representations of the correlators in the large-charge limit. Amusingly, the number of colors in these matrix models is formally taken to zero in the relevant limit.