  We develop a general method to calculate the exact time dependence of the
cumulants of the position of a tracer particle in a dense lattice gas of
hardcore particles. More precisely, we calculate the cumulant generating
function associated with the position of a tagged particle at arbitrary time,
and at leading order in the density of vacancies on the lattice. In particular,
our approach gives access to the short-time dynamics of the cumulants of the
tracer position -- a regime in which few results are known. The generality of
our approach is demonstrated by showing that it goes beyond the case of a
symmetric 1D random walk, and covers the important situations of (i) a biased
tracer; (ii) comb-like structures; and (iii) $d$-dimensional situations.
