In a crowded liquid, how likely is it that two particles stay unusually close to each other for a long time? Such rare events matter for processes like chemical reactions, aggregation, or ion pairing in electrolytes, yet they lie beyond the reach of standard tools. On the one hand, the theory of liquids describes the average structure through pair quantities, such as the pair correlation function or the cavity distribution function, but says little about their fluctuations. On the other hand, stochastic density functional theory, based on the Dean-Kawasaki equation, describes fluctuations, but usually relies on a linearization that fails for strong interactions and erases local structure. The goal of this internship is to bridge both approaches, by deriving and analyzing stochastic equations for the separation between pairs of particles. Path-integral methods could then give access to the statistics of time-integrated pair observables. The work will be mostly analytical, complemented by Brownian dynamics simulations.