Geodesics cross any pattern in first-passage percolation without any moment assumption and with possibly infinite passage times

Abstract

In first-passage percolation, one places nonnegative i.i.d. random variables ($T\left(e\right)$) on the edges of $Z^d$. A geodesic is an optimal path for the passage times $T\left(e\right)$. Consider a local property of the time environment. We call it a pattern. We investigate the number of times a geodesic crosses a translate of this pattern. When we assume that the common distribution of the passage times satisfies a suitable moment assumption, it is shown in [Antonin Jacquet. Geodesics in first-passage percolation cross any pattern, arXiv:2204.02021, 2023] that, apart from an event with exponentially small probability, this number is linear in the distance between the extremities of the geodesic. This paper completes this study by showing that this result remains true when we consider distributions with an unbounded support without any moment assumption or distributions with possibly infinite passage times. The techniques of proof differ from the preceding article and rely on a notion of penalized geodesic.