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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
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