Note on the maximal jump size in a continuum model of directed first passage percolation

Abstract

In this note, we consider the directed first passage percolation introduced in [F. Comets, R. Fukushima, S. Nakajima and N. Yoshida: Journal of Statistical Physics, 161-(3), 577–597 (2015)]. It is proved that the shortest path from the origin to the $n$-th hyperplane makes a jump larger than a positive power of $\log n$. Some numerical results are also provided, which indicates that the maximal jump size is much larger in a certain parameter region.