The behavior of long optimal paths in random environments has been studied in many models, like last/first passage percolations or random Hamilton—Jacobi equations. In various settings, these optimal paths have some kind of straightness property, which leads to the existence of asymptotic directions for such paths. Motivated by this, we consider a simpler setting where the control is eliminated and trajectories are giving by the integral curves of a random ergodic vector field, and ask how decorrelated the random environment should be to guarantee the existence of asymptotic directions. In this talk we will present a counterexample for mixing vector fields. This is joint work with Yuri Bakhtin.