Geodesics in first-passage percolation cross any pattern

In first-passage percolation, one places nonnegative i.i.d. random variables (T(epsilon)) on the edges of Zd. A geodesic is an optimal path for the passage times T(epsilon). Consider a local property of the time environment. We call it a pattern. We investigate the number of times a geodesic crosses a translation of this pattern. Under mild conditions, we show that, apart from an event with exponentially small probability, this number is linear in the distance between the endpoints.