  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.
