  Inflationary perturbations are approximately Gaussian and deviations from
Gaussianity are usually calculated using in-in perturbation theory. This
method, however, fails for unlikely events on the tail of the probability
distribution: in this regime non-Gaussianities are important and perturbation
theory breaks down for $|\zeta| \gtrsim |f_{\rm \scriptscriptstyle NL}|^{-1}$.
In this paper we show that this regime is amenable to a semiclassical
treatment, $\hbar \to 0$. In this limit the wavefunction of the Universe can be
calculated in saddle-point, corresponding to a resummation of all the
tree-level Witten diagrams. The saddle can be found by solving numerically the
classical (Euclidean) non-linear equations of motion, with prescribed boundary
conditions. We apply these ideas to a model with an inflaton self-interaction
$\propto \lambda \dot\zeta^4$. Numerical and analytical methods show that the
tail of the probability distribution of $\zeta$ goes as
$\exp(-\lambda^{-1/4}\zeta^{3/2})$, with a clear non-perturbative dependence on
the coupling. Our results are relevant for the calculation of the abundance of
primordial black holes.
