Subgaussian concentration and rates of convergence in directed polymers

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 $\sqrt{N\mathrm{/}\log N}$ . This is used to show that $$\mathbb{E}$[ \log Z_{N,\omega}]$differsfrom$N$ times the free energy by an amount which is also subgaussian (i.e. $o(\sqrt{N})$), specifically $O(\sqrt{\frac{N}{\log N}}\log \log N)$.