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For the two-sided Bernoulli initial condition with density $\rho_-$ (resp.$\rho_+$) to the left (resp. to the right), we study the distribution of atagged particle in the one dimensional symmetric simple exclusion process. Weobtain a formula for the moment generating function of the associated currentin terms of a Fredholm determinant. Our arguments are based on a combination oftechniques from integrable probability which have been developed recently forstudying the asymmetric exclusion process and a subsequent intricate symmetriclimit. An expression for the large deviation function of the tagged particleposition is obtained, including the case of the stationary measure with uniformdensity $\rho$.

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