Weighing the “heaviest” Polya urn

For the classical Polya urn model parametrized by exponent γγ, the limit distributions of the fraction of balls in each bin are simple and well known when γ<1γ<1 (“egalitarian”) and when γ>1γ>1 (“winner takes all”). In this note, we partially fill in the gap for γ=1γ=1, the critical point, by providing explicit analytical expressions for all the moments of the limit distribution of the fraction of balls in the bin with the most balls.