  We consider directed random polymers in $(d+1)$ dimensions with nearly gamma
i.i.d. disorder. We study the partition function $Z_{N,\omega}$ and establish
exponential concentration of $\log Z_{N,\omega}$ about its mean on the
subgaussian scale $\sqrt{N/\log N}$ . This is used to show that $\mathbb{E}[
\log Z_{N,\omega}]$ differs from $N$ times the free energy by an amount which
is also subgaussian (i.e. $o(\sqrt{N})$), specifically $O(\sqrt{\frac{N}{\log
N}}\log \log N)$.
