In this talk, I will describe how a large deviation principle can be used to follow the gravitational evolution of the large scale structure of matter in the Universe into the mildly nonlinear regime. This approach allows for analytical predictions for counts-in-cells statistics that are beyond what is commonly achievable via other statistics such as correlation functions. The idea is to determine the mean cosmic densities within concentric spheres and to study their joint statistics. This symmetry allows to use spherical collapse dynamics which leads to surprisingly accurate predictions where standard perturbative calculations usually break down. In particular, I will show results for the density and velocity statistics in concentric cells in comparison to state-of-the-art numerical simulations. Finally, I will discuss current efforts to extend the formalism to include primordial non-Gaussianity, bias and cylindrical cells and exploit its potential to constrain cosmology using future surveys.
