Approximation of subadditive functions and convergence rates in limiting-shape results

Abstract

For a nonnegative subadditive function $h$ on $\mathbb{Z}^d$, with limiting approximation $g(x) = \lim_n h(nx)/n$, it is of interest to obtain bounds on the discrepancy between $g(x)$ and $h(x)$, typically of order $|x|^{\nu}$ with $\nu < 1$. For certain subadditive $h(x)$, particularly those which are expectations associated with optimal random paths from 0 to $x$, in a somewhat standardized way a more natural and seemingly weaker property can be established: every $x$ is in a bounded multiple of the convex hull of the set of sites satisfying a similar bound. We show that this convex-hull property implies the desired bound for all $x$. Applications include rates of convergence in limiting-shape results for first-passage percolation (standard and oriented) and longest common subsequences and bounds on the error in the exponential-decay approximation to the off-axis connectivity function for subcritical Bernoulli bond percolation on the integer lattice.