Timeline for Existence of a bigeodesic in last passage percolation is $0$-$1$ event
Current License: CC BY-SA 4.0
11 events
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| Nov 25, 2020 at 0:07 | history | edited | YCor | CC BY-SA 4.0 |
formatting, unabbreviated
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| Nov 25, 2020 at 0:00 | comment | added | Raghav | Sorry for the delay. I have rolled backed it to the original version. Thanks! | |
| Nov 24, 2020 at 23:59 | history | rollback | Raghav |
Rollback to Revision 1
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| Nov 24, 2020 at 4:34 | comment | added | Yuval Peres | 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. | |
| Nov 23, 2020 at 6:57 | comment | added | Anthony Quas | It is considered bad form to remove the old parts of the question so that the comments no longer make sense. It's OK to make edits that are minor changes to the question, but readers should still be able to see the original post that the comments refer to. | |
| Nov 23, 2020 at 6:30 | history | edited | Raghav | CC BY-SA 4.0 |
Removed two questions!
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| Nov 23, 2020 at 4:30 | comment | added | Anthony Quas | 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. | |
| Nov 23, 2020 at 4:20 | comment | added | Raghav | 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. | |
| Nov 23, 2020 at 4:16 | comment | added | Raghav | @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. | |
| Nov 23, 2020 at 3:51 | comment | added | Anthony Quas | (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). | |
| Nov 23, 2020 at 0:51 | history | asked | Raghav | CC BY-SA 4.0 |