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@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?
@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 $d\geq4$?
@IgorBelegradek, as far as I understood from the construction of the orthonormal frame bundle, there is a 1-1 correspondence between paths in $\mathbb{R}^2$ and paths in $\mathbb{S}^2$ in an isometric way, which doesn't require charts. The question is how to realize this 1-1 map explicitly, thinking extrinsically of the sphere embedded in $\mathbb{R}^3$, starting from a curve in $\mathbb{R}^2$. In your comment you fix two curves on the two spaces, but it's not clear (to me) how to relate the two
