  We study large charge sectors in the $O(N)$ model in $6-\epsilon $
dimensions. For $4<d<6$, in perturbation theory, the quartic $O(N)$ theory has
a UV stable fixed point at large $N$. It was recently argued that this fixed
point can be described in terms of an IR fixed point of a cubic $O(N)$ model.
By considering a double scaling limit of large charge and weak couplings, we
compute two-point and all "extremal" higher-point correlation functions for
large charge operators and find a precise equivalence between both pictures.
Instanton instabilities are found to be exponentially suppressed at large
charge. We also consider correlation function of $U(1)$-invariant meson
operators in the $O(2N)\supset U(1)\times SU(N)$ theory, as a first step
towards tests of (higher spin) $AdS/CFT$.
