A Hsu-Robbins-Erd\H{o}s strong law in first-passage percolation

TL;DR

This paper establishes a precise relation between polynomial moments and tail decay in first-passage percolation, strengthening the shape theorem without requiring higher-order moments, extending classical strong law results.

Contribution

It derives a new relation connecting moments and tail decay in first-passage percolation, enhancing the shape theorem without higher-order moment assumptions.

Findings

Established a relation between moments and tail decay in first-passage percolation.

Strengthened the shape theorem without higher-order moment conditions.

Extended Hsu-Robbins-Erdős strong law to a new context.

Abstract

Large deviations in the context of first-passage percolation was first studied in the early 1980s by Grimmett and Kesten, and has since been revisited in a variety of studies. However, none of these studies provides a precise relation between the existence of moments of polynomial order and the decay of probability tails. Such a relation is derived in this paper, and is used to strengthen the conclusion of the shape theorem. In contrast to its one-dimensional counterpart - the Hsu-Robbins-Erd\H{o}s strong law - this strengthening is obtained without imposing a higher-order moment condition.