Timeline for Infinite clusters for loopless percolation
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 1, 2023 at 22:15 | answer | added | tmh | timeline score: 3 | |
| Sep 10, 2022 at 11:25 | comment | added | Viktor B | It seems that the definition needs to be expanded because $\phi(\omega) = 0$ whenever $E$ is infinite. | |
| Sep 10, 2022 at 8:39 | comment | added | tmh | In particular I don't think we know that there is a well-defined "free Gibbs measure" or "wired Gibbs measure", just that translation invariant Gibbs measures always exist (by general abstract nonsense). For e.g. the random cluster model with q >= 1 the existence and extremality of the wired and free measures follows from FKG. The arboreal gas should probably be negatively correlated (which suggests that free and wired should be extremal) but this is not known. | |
| Sep 10, 2022 at 8:34 | comment | added | tmh | Currently we don't know anything about the structure of the Gibbs measures. It seems plausible that there is only one Gibbs measure in the supercritical regime, but the existing techniques don't really tell us anything AFAIK. | |
| Sep 10, 2022 at 7:18 | comment | added | PeaBrane | @tmh that's very interesting! If I may ask, when you say "any translation invariant Gibbs measure", do you mean the two extremal ones induced by the free and wired b.c? Or perhaps there's some non-trivial multiplicity of extremal states for these models. | |
| Sep 10, 2022 at 3:20 | comment | added | tmh | Regarding your second question, I have a forthcoming paper with Noah Halberstam where we prove that there is always at most one infinite cluster in three and four dimensions (in any translation invariant Gibbs measure for the model). Our arguments also give strong heuristic evidence that there should be infinitely many infinite clusters in the supercritical phase in five dimensions and higher (as there is in the USF, which is the p=1 limit). | |
| Sep 9, 2022 at 14:34 | history | became hot network question | |||
| Sep 9, 2022 at 14:05 | vote | accept | PeaBrane | ||
| Sep 9, 2022 at 9:51 | answer | added | James Martin | timeline score: 7 | |
| Sep 9, 2022 at 6:37 | history | edited | PeaBrane | CC BY-SA 4.0 |
added 80 characters in body
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| Sep 9, 2022 at 6:31 | history | asked | PeaBrane | CC BY-SA 4.0 |