We study the correlation functions of Coulomb branch operators of four-dimensional $mathcal{N} = 2$ Superconformal Field Theories (SCFTs). We focus on rank-one theories, such as the SU(2) gauge theory with four fundamental hypermultiplets. Extremal correlation functions, involving exactly one anti-chiral operator, are perhaps the simplest nontrivial correlation functions in four-dimensional Quantum Field Theory. We show that the large charge limit of extremal correlators is captured by a dual description which is a chiral random matrix model of the Wishart-Laguerre type. This gives an analytic handle on the physics in some particular excited states. In the limit of large random matrices we find the physics of a non-relativistic axion-dilaton effective theory. The random matrix model also admits a t Hooft expansion in which the matrix is taken to be large and simultaneously the coupling is taken to zero. This explains why the extremal correlators of SU(2) gauge theory obey a nontrivial double scaling limit in states of large charge. We give an exact solution for the first two orders in the t Hooft expansion of the random matrix model and compare with expectations from effective field theory, previous weak coupling results, and we analyze the non-perturbative terms in the strong t Hooft coupling limit. Finally, we apply the random matrix theory techniques to study extremal correlators in rank-1 Argyres-Douglas theories. We compare our results with effective field theory and with some available numerical bootstrap bounds.
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