  Diffusion of impenetrable particles in a crowded one-dimensional channel is
referred as the single file diffusion. The particles do not pass each other and
the displacement of each individual particle is sub-diffusive. We analyse a
simple realization of this single file diffusion problem where one dimensional
Brownian point particles interact only by hard-core repulsion. We show that the
large deviation function which characterizes the displacement of a tracer at
large time can be computed via a mapping to a problem of non-interacting
Brownian particles. We confirm recently obtained results of the one time
distribution of the displacement and show how to extend them to the multi-time
correlations. The probability distribution of the tracer position depends on
whether we take annealed or quenched averages. In the quenched case we notice
an exact relation between the distribution of the tracer and the distribution
of the current. This relation is in fact much more general and would be valid
for arbitrary single file diffusion. It allows in particular to get the full
statistics of the tracer position for the symmetric simple exclusion process
(SSEP) at density 1/2 in the quenched case.
