Timeline for Mathematical construction of $\phi^4$ Euclidean field theory
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Feb 6, 2021 at 16:03 | comment | added | Martin Hairer | @PPR What would be the precise meaning of the statement "$\Phi^4_4$ doesn't exist"? Yes, there are reasonable looking approximations of $\Phi^4$ to which A-DC doesn't apply and this will be always be true, whatever more general statement of this form they come up with. I don't think anyone would expect any other approximation to behave in a substantially different way. | |
| Feb 6, 2021 at 15:39 | comment | added | PPR | @MartinHairer, Thanks. My understanding was that what A.-D.C. prove in $d=4$ is that any scaling limit of a lattice $\Phi^4$ theory converges to a Gaussian limit. Is it clear that that implies there is no $\Phi^4$ measure in $d=4$, rather than just that if $\Phi^4_4$ exists, it cannot be obtained via a scaling limit of a discrete theory? | |
| Feb 6, 2021 at 12:18 | comment | added | Martin Hairer | @AbdelmalekAbdesselam Yes, that's the one. | |
| Feb 6, 2021 at 8:26 | comment | added | Martin Hairer | @PPR There exists no $\Phi^4$ measure in $d\ge 4$, see this recent paper by Aizenman and Duminil-Copin. | |
| Feb 6, 2021 at 0:34 | comment | added | PPR | @MartinHairer, thanks for the reference to the proof. Is it obvious from the fact the $d=3$ measure is singular w.r.t. the GFF that the same is true also for $d>3$? If not, are there analogous proofs for $d\geq4$? | |
| Feb 6, 2021 at 0:31 | vote | accept | PPR | ||
| Feb 6, 2021 at 0:16 | comment | added | Abdelmalek Abdesselam | @MartinHairer: I guess that's what you showed us at the Imperial College conference in 2019. Thanks for posting it, and making it available. | |
| Feb 5, 2021 at 21:43 | comment | added | Martin Hairer | A sketch of a slightly different proof of that singularity can be found here (this is the unpublished proof mentioned on p.3 of the above preprint). | |
| Feb 5, 2021 at 18:28 | comment | added | Abdelmalek Abdesselam | For a proof of the singularity of measures mentioned by Martin you can see, e.g., arxiv.org/abs/2004.01513 | |
| Feb 5, 2021 at 18:28 | comment | added | Abdelmalek Abdesselam | Related: mathoverflow.net/questions/260854/… the part at the end. As Martin said, trying to use the free field as your underlying fixed (anchored) probability space and construct (even by a limiting procedure) $\int \phi^4$ as a functional of that free field, is okay in 2d and finite volume but will not work in 3d. There you need to see the problem as that of weak convergence of a sequence of cut-off measures (fixed measurable space, but ton of probability spaces). | |
| Feb 5, 2021 at 10:14 | answer | added | Martin Hairer | timeline score: 12 | |
| Feb 5, 2021 at 4:09 | history | asked | PPR | CC BY-SA 4.0 |