  The Symmetric Exclusion Process (SEP), in which particles hop symmetrically
on a discrete line with hard-core constraints, is a paradigmatic model of
subdiffusion in confined systems. This anomalous behavior is a direct
consequence of strong spatial correlations induced by the requirement that the
particles cannot overtake each other. Even if this fact has been recognised
qualitatively for a long time, up to now there is no full quantitative
determination of these correlations. Here we study the joint probability
distribution of an arbitrary number of tagged particles in the SEP. We
determine analytically the large time limit of all cumulants for an arbitrary
density of particles, and their full dynamics in the high density limit. In
this limit, we unveil a universal scaling form shared by the cumulants and
obtain the time-dependent large deviation function of the problem.
