In the Last passage percolation in $\mathbb{Z}^2$ (with iid exponential weight), an infinite up/right path $\gamma$ is called a bigeodesic if for every pair of vertex $x, y\in \gamma$ the restriction of $\gamma$ from $x$ to $y$ is a geodesic. I want to show that the event $E=\{\text{there exists a bigeodesic}\}$ is a tail event. I know an easy proof using the ergodicity and translation invariance.
But I want a more down to earth approach. My idea is that we have a collection of i.i.d random variable $X_v, v\in \mathbb{Z}^2,$ and I want to say that the existence of bigeodesic does not depend upon any finite collection of random variables. To do this, I have a 'proof' as follows:
Let $B_n$ be the cube of the size of $n$ in $\mathbb{Z}^2.$ I want to show that the existence of bigeodesic is independent of $\{X_v: v\in B_n\}.$ My argument is as follows: $E_0:=\{\text{there exists a bigeodesic not intersecting } $B_n$\}$ is clearly independent of $B_n.$ The event that there exists a bigeodesic passing through $B_n$ is a union of two events. The two events being the existence of a semi-infinite geodesic ending at the left-bottom 'boundary' of $B_n$ and a semi-infinite geodesic starting from the top-right boundary of $B_n.$ These two events are independent of $B_n.$ I feel that this argument is not rigorous enough. Can anyone help me write the argument rigorously?