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On the bottom of page two of This paper, the authors remark the following:

'...by translation invariance and ergodicity, we know that existence of a bigeodesic is a $0−1$ event and hence it follows that if almost surely bigeodesics exist, then with positive probability there must exist bigeodesics passing through the origin.

I have a few questions about this remark.

(A) In a last passage percolation, we always have trivial bigeodesics (horizontal and vertical lines) and therefore it follows that the existence of a bigeodesic is an almost sure event. Am I missing something here? Or the above remark is for the existence of a non-trivial bigeodesic?

(B) What do they mean by 'translation invariance and ergodicity'? I mean what exactly is the argument here?

(C) It intuitively seems reasonable to believe that the existence of bigeodesic should not be affected by finitely many vertices. In other words, one should be able to say that the existence of a bigeodesic (or a non-trivial bigeodesic) is a $0$-$1$ event as one does in Kolmogorov's $0$-$1$ law. I tried it but could not make a proper argument.

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    $\begingroup$ (A) yes they mean the existence of a non-trivial bigeodesic is a translation-invariant event, so if measure 0 or 1; (B) the definition of ergodicity is that any event that is invariant under any translation by all elements of $\mathbb Z^2$ is of measure 0 or 1. It’s a standard argument that the model the authors are considering is ergodic. (A) follows immediately (since it’s easy to see the configurations containing a non-trivial bigeodesic are invariant; and not too hard to see that they’re measurable). $\endgroup$
    – Anthony Quas
    Commented Nov 23, 2020 at 3:51
  • $\begingroup$ @AnthonyQuas thanks for the comment. If I understand you correctly, you mean if $\gamma$ is a bigeodesic in one realization then translating everything gives another relaization with a bigeodesic $\gamma’=\gamma+k$. And this is what meant by the translation invariance of the event? At this point the authors use the fact that the model is ergodic and hence the above event is a trivial event. $\endgroup$
    – Raghav
    Commented Nov 23, 2020 at 4:16
  • $\begingroup$ Since I do not have a background in Ergodic theory but I have seen the concept of trivial events in the context of Kolmogorov zero-one law, I would like to show that the event $E$ that there exists a bigeodesic is a tail event, where sigma algebra that we are considering is generated by the passage times at each vertex. To this end, I tried something like considering a box $B_n$ and then trying to show that the event $E$ is independent of $B_n.$ But I did not succeed, any pointer in this direction would be really helpful. $\endgroup$
    – Raghav
    Commented Nov 23, 2020 at 4:20
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    $\begingroup$ Here is the argument. Let $S$ be the set of configurations with a bigeodesic. It’s not clear to me that $S$ is a tail event. However suppose $\omega\in S$. Then $\tau_v(\omega)\in S$ where $\tau_v$ is translation by $v$. It follows that $S$ is invariant set under the group of translations. Since the probability measure is known to be ergodic under the group of translations, any translation-invariant measurable set is of measure 0 or 1. $\endgroup$
    – Anthony Quas
    Commented Nov 23, 2020 at 4:30
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    $\begingroup$ To the original proposer: Please restore parts (A) and (B) of your question so that the comments make sense. Changing a question after it is answered or commented on is considered bad form, except for minor clarifications. $\endgroup$
    – Yuval Peres
    Commented Nov 24, 2020 at 4:34

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