The statistics of primordial scalar perturbations play a crucial role in providing phenomenological constraints for new inflationary models. In particular, the probability of unlikely large scalar perturbations (leading to primordial black hole formation) and the concurrent GW backgrounds are very sensitive to the tail of the PDF of primordial curvature perturbations. We analyse such tails in inflationary models featuring an ultra-slow roll phase, known to enhance both the amplitude and non-Gaussianity of curvature perturbations at small scales. We compare two methods to compute the PDF, relying both on classical non-linear techniques ($\delta N$ formalism) and the stochastic inflation paradigm, and explain their agreement in this kind of scenarios. Finally, we discuss the importance of corrections arising from the inflaton self-interactions to standard calculations of the PDF.

Based on 2406.02417 and upcoming work with G. Ballesteros, J. Gambín Egea, T. Konstandin, M. Pierre, J. Rey.