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

TL;DR

This paper investigates the maximal jump size in a continuum model of directed first passage percolation, proving it exceeds a small power of log n and providing numerical evidence of larger jumps in certain parameter regions.

Contribution

It establishes a lower bound on the maximal jump size in the model and offers numerical insights into parameter-dependent jump behaviors.

Findings

Maximal jump size exceeds a small power of log n

Numerical results suggest larger jumps in specific parameter regions

Provides theoretical and numerical analysis of jump sizes

Abstract

In this note, we study 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 small 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.