Tagged particle in single-file diffusion with arbitrary initial conditions

We compute the full probability distribution of the positions of a tagged particle exactly for the given arbitrary initial positions of the particles, and for general single-particle propagators. We consider the thermodynamic limit of our exact expressions in quenched and annealed settings. For a particular class of single-particle propagators, the exact formula is expressed in a simple integral form in the quenched case whereas in the annealed case, it is expressed as a simple combination of Bessel functions. In particular, we focus on the step and the power-law initial configurations. In the former case, a drift is induced even when the one-particle propagators are symmetric. On the other hand, in the later case the scaling of the cumulants of the position of the tracer differs from the uniform case. We provide numerical verifications of our results.