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Tripartite information at long distances

We compute the leading term of the tripartite information at long distances for three spheres in a CFT. This falls as $r^{-6Δ}$, where $r$ is the typical distance between the spheres, and $Δ$, 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 $r^{-6Δ_f}$. In particular, lattice calculations suggest a leading scaling $ r^{-(6Δ_f+1)}$. For free fermions in three dimensions, we show that mutual information is also non-monogamous in the long-distance regime.