Rate of convergence of the mean for sub-additive ergodic sequences

For sub-additive ergodic processes $\{X_{m,n}\}$ with weak dependence, we analyze the rate of convergence of $\mathbb{E}{X}_{0,n}\mathrm{/}n$ to its limit $g$. We define an exponent $\gamma$ given roughly by $$\mathbb{E}$X_{0,n} \sim ng + n^\gamma$,and,assumingexistenceofafluctuationexponent$\chi$thatgives$$\mathrm{V}\mathrm{a}\mathrm{r}$~X_{0,n} \sim n^{2\chi}$,weprovidealowerboundfor$\gamma$ of the form $\gamma \ge \chi$. The main requirement is that $\chi \mathrm{\ne }1\mathrm{/}2$. In the case $\chi =1\mathrm{/}2$ and under the assumption $$\mathrm{V}\mathrm{a}\mathrm{r}$~X_{0,n} = O(n/(\log n)^\beta)$forsome$\beta>0$,weprove$\gamma \geq \chi - c(\beta)$fora$\beta$-dependentconstant$c(\beta)$. These results show in particular that non-diffusive fluctuations are associated to non-trivial $\gamma$. Various models, including first-passage percolation, directed polymers, the minimum of a branching random walk and bin packing, fall into our general framework, and the results apply assuming $\chi$ exists. In the case of first-passage percolation in ${\mathbb{Z}}^d$, we provide a version of $\gamma \ge -1\mathrm{/}2$ without assuming existence of $\chi$.