  The 1/2-BPS Wilson loop in $\mathcal{N}=4$ supersymmetric Yang-Mills theory
is an important and well-studied example of conformal defect. In particular,
much work has been done for the correlation functions of operator insertions on
the Wilson loop in the fundamental representation. In this paper, we extend
such analyses to Wilson loops in the large-rank symmetric and antisymmetric
representations, which correspond to probe D3 and D5 branes with $AdS_2 \times
S^2$ and $AdS_2 \times S^4$ worldvolume geometries, ending at the $AdS_5$
boundary along a one-dimensional contour. We first compute the correlation
functions of protected scalar insertions from supersymmetric localization, and
obtain a representation in terms of multiple integrals that are similar to the
eigenvalue integrals of the random matrix, but with important differences.
Using ideas from the Fermi Gas formalism and the Clustering method, we evaluate
their large $N$ limit exactly as a function of the 't Hooft coupling. The
results are given by simple integrals of polynomials that resemble the
$Q$-functions of the Quantum Spectral Curve, with integration measures
depending on the number of insertions. Next, we study the correlation functions
of fluctuations on the probe D3 and D5 branes in AdS. We compute a selection of
three- and four-point functions from perturbation theory on the D-branes, and
show that they agree with the results of localization when restricted to
supersymmetric kinematics. We also explain how the difference of the internal
geometries of the D3 and D5 branes manifests itself in the localization
computation.
