The four-point function of determinant operators in $\mathcal{N}=4$ SYM

We calculate the four-point function of $1/2$-BPS determinant operators in $\mathcal{N}=4$ SYM at next-to-leading order at weak coupling. We use two complementary methods recently developed for a class of determinant three-point functions: one is based on Feynman diagrams and it extracts perturbative data at finite $N$, while the other one expresses a generic correlator of determinants as the zero-dimensional integral over an auxiliary matrix field. We generalise the latter approach to calculate one-loop corrections and we solve the four-point function in a semi-classical approach at large $N$. The results allow to comment on the order of the phase transition that the four-point function is expected to exhibit in an exact integrability-based description.