Distribution of a tagged particle position in the one-dimensional symmetric simple exclusion process with two-sided Bernoulli initial condition

For the two-sided Bernoulli initial condition with density $ρ_-$ (resp. $ρ_+$) to the left (resp. to the right), we study the distribution of a tagged particle in the one dimensional symmetric simple exclusion process. We obtain a formula for the moment generating function of the associated current in terms of a Fredholm determinant. Our arguments are based on a combination of techniques from integrable probability which have been developed recently for studying the asymmetric exclusion process and a subsequent intricate symmetric limit. An expression for the large deviation function of the tagged particle position is obtained, including the case of the stationary measure with uniform density $ρ$.