Probability distributions of the order parameter of the $O(N)$ model

We study the probability distribution function (PDF) of the order parameter of the three-dimensional $O(N)$ model at criticality using the functional renormalisation group. For this purpose, we generalize the method introduced in [Balog et al., Phys. Rev. Lett. {\bf 129}, 210602 (2022)] to the $O(N)$ model. We study the large $N$ limit, as well as the cases $N=2$ and $N=3$ at the level of the Local Potential Approximation (LPA), and compare our results to Monte Carlo simulations. We compute the entire family of universal scaling functions, obtained in the limit where the system size $L$ and the correlation length of the infinite system ${\xi }_{\mathrm{\infty }}$ diverge, with the ratio $\zeta =L\mathrm{/}{\xi }_{\mathrm{\infty }}$ constant. We also generalize our results to the approach of criticality from the low-temperature phase where another infinite family of universal PDF exists. We find that the LPA describes very well the functional form of the family of PDFs, once we correct for a global amplitude of the (logarithm of the) PDF and of $\zeta$.