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January 1997 Approximation of subadditive functions and convergence rates in limiting-shape results
Kenneth S. Alexander
Ann. Probab. 25(1): 30-55 (January 1997). DOI: 10.1214/aop/1024404277


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.


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Kenneth S. Alexander. "Approximation of subadditive functions and convergence rates in limiting-shape results." Ann. Probab. 25 (1) 30 - 55, January 1997.


Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0882.60090
MathSciNet: MR1428498
Digital Object Identifier: 10.1214/aop/1024404277

Primary: 60K35
Secondary: 41A25 , 60C05 , 82B43

Keywords: Connectivity function , First-passage percolation , Longest common subsequence , oriencted first-passage percolation , subadditivity‎

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 1 • January 1997
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