A Hsu-Robbins-Erdős strong law in first-passage percolation. (English) Zbl 1321.60199

In this paper, a systematic study of the regime for polynomial decay (RPD) of the probability tails is carried out. A precise characterization of the RPD in terms of a moment condition is derived, and, as a consequence, the result improves the statement of the shape theorem without strengthening its hypothesis. In general, the obtained results strengthen earlier strong laws in first-passage percolation, in particular the shape theorem due to D. Richardson [Proc. Camb. Philos. Soc. 74, 515–528 (1973; Zbl 0295.62094)] and J. T. Cox and R. Durrett [Ann. Probab. 9, 583–603 (1981; Zbl 0462.60012)], which states the precise conditions under which the set of points in \(Z^d\) within distance \(t\) from the origin, rescaled by \(t\), converges to a deterministic compact or convex set.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
60F15 Strong limit theorems
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