Strict inequality between the time constants of first-passage percolation and directed first-passage percolation

In the models of first-passage percolation and directed first-passage percolation on $\mathbb{Z}^d$, we consider a family of i.i.d. random variables indexed by the set of edges of the graph, called passage times. For every vertex $x \in \mathbb{Z}^d$ with nonnegative coordinates, we denote by $t(0,x)$ the shortest passage time to go from $0$ to $x$ and by $\vec t(0,x)$ the shortest passage time to go from $0$ to $x$ following a directed path. Under some assumptions, it is known that for every $x \in \mathbb{R}^d$ with nonnegative coordinates, $t(0,\lfloor nx \rfloor)/n$ converges to a constant $\mu(x)$ and that $\vec t(0,\lfloor nx \rfloor)/n$ converges to a constant $\vec\mu(x)$. With these definitions, we immediately get that $\mu(x) \le \vec{\mu}(x)$. In this paper, we get the strict inequality $\mu(x) < \vec\mu(x)$ as a consequence of a new exponential bound for the comparison of $t(0,x)$ and $\vec{t}(0,x)$ when $\|x\|$ goes to $\infty$. This exponential bound is itself based on a lower bound on the number of edges of geodesics in first-passage percolation (where geodesics are paths with minimal passage time).