Clustering of primordial black holes from quantum diffusion during inflation

We study how large fluctuations are spatially correlated in the presence of quantum diffusion during inflation. This is done by computing real-space correlation functions in the stochastic-$\delta N$ formalism. We first derive an exact description of physical distances as measured by a local observer at the end of inflation, improving on previous works. Our approach is based on recursive algorithmic methods that consistently include volume-weighting effects. We then propose a "large-volume'' approximation under which calculations can be done using first-passage time analysis only, and from which a new formula for the power spectrum in stochastic inflation is derived. We then study the full two-point statistics of the curvature perturbation. Due to the presence of exponential tails, we find that the joint distribution of large fluctuations is of the form $P(\zeta_{R_1}, \zeta_{R_2}) = F(R_1,R_2,r) P(\zeta_{R_1})P( \zeta_{R_2})$, where $\zeta_{R_1}$ and $\zeta_{R_2}$ denote the curvature perturbation coarse-grained at radii $R_1$ and $R_2$, around two spatial points distant by $r$. This implies that, on the tail, the reduced correlation function, defined as $P(\zeta_{R_1}>\zeta_{\rm{c}}, \zeta_{R_2}>\zeta_{\rm{c}})/[P(\zeta_{R_1}>\zeta_{\rm{c}}) P(\zeta_{R_2}>\zeta_{\rm{c}})]-1$, is independent of the threshold value $\zeta_{\rm{c}}$. This contrasts with Gaussian statistics where the same quantity strongly decays with $\zeta_{\rm{c}}$, and shows the existence of a universal clustering profile for all structures forming in the exponential tails. Structures forming in the intermediate (i.e. not yet exponential) tails may feature different, model-dependent behaviours.