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

TL;DR

This paper investigates the rate at which the expected value of sub-additive ergodic processes converges to its limit, establishing bounds related to fluctuation exponents and applying to models like first-passage percolation and directed polymers.

Contribution

It introduces bounds on the convergence rate exponent for sub-additive ergodic sequences, linking it to fluctuation exponents and extending results to various models.

Findings

Established lower bounds for the convergence rate exponent b3 based on fluctuation exponent b3 .

Connected non-diffusive fluctuations to non-trivial convergence rates.

Applied results to models including first-passage percolation, directed polymers, and branching random walks.

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 $\gamma \geq \chi$. The main requirement is that $\chi \neq 1/2$. 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 $\gamma \geq \chi - c(\beta)$ 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 $\gamma \geq -1/2$ without assuming existence of $\chi$.