## 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
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### References:

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