The Annals of Probability
- Ann. Probab.
- Volume 25, Number 1 (1997), 30-55.
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.
Article information
Source
Ann. Probab., Volume 25, Number 1 (1997), 30-55.
Dates
First available in Project Euclid: 18 June 2002
Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404277
Digital Object Identifier
doi:10.1214/aop/1024404277
Mathematical Reviews number (MathSciNet)
MR1428498
Zentralblatt MATH identifier
0882.60090
Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35] 41A25: Rate of convergence, degree of approximation 60C05: Combinatorial probability
Keywords
subadditivity first-passage percolation longest common subsequence oriencted first-passage percolation connectivity function
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
