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**Approximation of subadditive functions and convergence rates in limiting-shape results.**
*(English)*
Zbl 0882.60090

Summary: 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.

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B43 | Percolation |

41A25 | Rate of convergence, degree of approximation |

60C05 | Combinatorial probability |

### Keywords:

subadditivity; first-passage percolation; longest common subsequence; oriented first-passage percolation; connectivity function
Full Text:
DOI

### References:

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