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

Abstract

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 $\gamma$ given roughly by $\mathbb{E}{X}_{0,n}\sim ng+n^{\gamma }$, and, assuming existence of a fluctuation exponent $\chi$ that gives $\mathrm{Var}{X}_{0,n}\sim n^{2\chi }$, we provide a lower bound for $\gamma$ of the form $\gamma \ge \chi$. The main requirement is that $\chi \ne 1/2$. In the case $\chi =1/2$ and under the assumption $\mathrm{Var}{X}_{0,n}=O(n/(\log n{)}^{\beta })$ for some $\beta >0$, we prove $\gamma \ge \chi -c(\beta )$ for a $\beta$-dependent constant $c(\beta )$. 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 $\gamma \ge -1/2$ without assuming existence of $\chi$.

arXiv comment41 pages, 2 figures. Expanded introduction and added a proof sketch