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 $\mathbb{E}X_{0,n}/n$ to its limit $g$. We define an exponent $γ$ given roughly by $\mathbb{E}X_{0,n} \sim ng + n^γ$, and, assuming existence of a fluctuation exponent $χ$ that gives $\mathrm{Var}~X_{0,n} \sim n^{2χ}$, we provide a lower bound for $γ$ of the form $γ\geq χ$. The main requirement is that $χ\neq 1/2$. In the case $χ=1/2$ and under the assumption $\mathrm{Var}~X_{0,n} = O(n/(\log n)^β)$ for some $β>0$, we prove $γ\geq χ- c(β)$ for a $β$-dependent constant $c(β)$. These results show in particular that non-diffusive fluctuations are associated to non-trivial $γ$. 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 $χ$ exists. In the case of first-passage percolation in $\mathbb Z^d$, we provide a version of $γ\geq -1/2$ without assuming existence of $χ$.