  We study correlation functions on the Coulomb branch of planar $\mathcal{N} =
4$ super-Yang- Mills theory (SYM), and their relationship with integrability,
the operator product expansion (OPE), the sum rule, the large charge expansion,
and holography. First, we compute one-point functions of arbitrary scalar
operators at weak coupling and derive a compact spin-chain representation. We
next study the two-point functions of chiral primaries at one loop and find
that the radius of convergence of OPE in position space is infinite. We
estimate the asymptotic growth of the OPE data based on this finding. Finally,
we propose a concrete nonperturbative formula that connects the correlation
functions on the Coulomb branch with the correlation functions with large
charge insertions at the conformal point and provide a holographic
interpretation based on a large D3-brane in AdS. The formula extends the known
connection between the large charge sector and the Coulomb branch for rank-1
theories to the large $N$ limit.
