  In this paper, we consider four integrable models of directed polymers for
which the free energy is known to exhibit KPZ fluctuations. A common framework
for the analysis of these models was introduced in our recent work on the
O'Connell-Yor polymer.
  We derive estimates for the central moments of the partition function, of any
order, on the near-optimal scale $N^{1/3+\epsilon}$, using an iterative method.
Among the innovations exploiting the invariant structure, we develop formulas
for correlations between functions of the free energy and the boundary weights
that replace the Gaussian integration by parts appearing in the analysis of the
O'Connell-Yor case.
