$1/2$ BPS Structure Constants and Random Matrices

We study three point functions of half BPS operators in $\mathcal{N}=4$ super Yang-Mills theory focusing on correltors of two of the operators with dimension of order $\Delta\sim N^2$ and a light single trace operator. These describe vacuum expectation values of type IIB supergravity modes in LLM backgrounds that do not necessarily preserve the same symmetries as the background solution. We propose a class of complex matrix models that fully capture the combinatorics of the problem, and describe their solution in the large $N$ limit. In simple regimes when the dual description is in terms of widely separated condensates of giant gravitons we find that the models are solvable in the large $N$ and can be approximated by unitary Jacobi ensembles; we describe how these distributions are reproduced in the dual bubbling geometry picture for large droplets. In the case of two eigenvalue droplets the model is exactly solvable at finite $N$. As a result we compute all half-BPS structure constants of heavy-heavy-light type, and reproduce the formulas found via holographic renormalization in the large $N$ limit. We also comment on structure constants of three heavy operators.