  We show that the amplitude of the finite size correction term for the $n$th
moment of the partition function, for randomly interacting directed polymers,
diverges (on the high temperature side) as $(n_c - n)^{-r}$, as a critical
moment $n_c$ is approached. The exponent $r$ is independent of temperature but
does depend on the effective dimensionality. There is no such divergence on the
low temperature side ($n>n_c)$.
