## The Annals of Probability

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

Kenneth S. Alexander

#### 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.

#### Article information

Source
Ann. Probab., Volume 25, Number 1 (1997), 30-55.

Dates
First available in Project Euclid: 18 June 2002

https://projecteuclid.org/euclid.aop/1024404277

Digital Object Identifier
doi:10.1214/aop/1024404277

Mathematical Reviews number (MathSciNet)
MR1428498

Zentralblatt MATH identifier
0882.60090

#### Citation

Alexander, Kenneth S. Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 (1997), no. 1, 30--55. doi:10.1214/aop/1024404277. https://projecteuclid.org/euclid.aop/1024404277

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