| Summary: | We compute the leading term of the tripartite information at long distances
for three spheres in a CFT. This falls as $r^{-6\Delta}$, where $r$ is the
typical distance between the spheres, and $\Delta$, the lowest primary field
dimension. The coefficient turns out to be a combination of terms coming from
the two- and three-point functions and depends on the OPE coefficient of the
field. We check the result with three-dimensional free scalars in the lattice
finding excellent agreement. When the lowest-dimensional field is a scalar, we
find that the mutual information can be monogamous only for quite large OPE
coefficients, far away from a perturbative regime. When the lowest-dimensional
primary is a fermion, we argue that the scaling must always be faster than
$r^{-6\Delta_f}$. In particular, lattice calculations suggest a leading scaling
$ r^{-(6\Delta_f+1)}$. For free fermions in three dimensions, we show that
mutual information is also non-monogamous in the long-distance regime.
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