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In first-passage percolation, one places nonnegative i.i.d. random variables(T(e)) on the edges of Z^d. A geodesic is an optimal path for the passage timesT(e). 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 thispattern. When we assume that the common distribution of the passage timessatisfies 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, thisnumber is linear in the distance between the extremities of the geodesic. Thispaper completes this study by showing that this result remains true when weconsider distributions with an unbounded support without any moment assumptionor distributions with possibly infinite passage times. The techniques of proofdiffer from the preceding article and rely on a notion of penalized geodesic.

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