In first passage percolation in $d$ dimensions, when the weights of the edges are i.i.d and distributed according to an atomless distribution $F$, it seems that for any $x,y \in \Bbb{Z}^d$, there exists a unique geodesic.
For existence, I think that we can proceed as follows :
Take any $x,y \in \Bbb{Z}^d$ and $\gamma : x \rightarrow y$ any path. As $F$ is atomless, in particular $F(\{x\}) = 0$ and thus $\underset{n \rightarrow \infty}{\lim} T(x,\partial B_n) = \infty$, where $B_n$ is the box of size $n$. Given this, we can always choose $n \in \Bbb{N}$ large enough so that $T(x, \partial B_n) \geq T(\gamma)$. Now take any path $\gamma'$ starting at $x$, exiting $B_n$ and finishing at $y$. We have $$T(\gamma') \geq T(x, \partial B_n) > T(\gamma)$$ We conclude that $$T(x,y) = \inf\{T(\gamma), \gamma : x \rightarrow y \} = \inf\{T(\gamma), \gamma : x \rightarrow y, \gamma \subset B_n\}$$ An infimum over a finite set of paths must be attained, hence there exists a geodesic $x \rightarrow y$.
Now, for the unicity part, I don't know how to proceed.
Thank you in advance for your help.