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- W2401672900 abstract "In Liouville quantum gravity (or 2d-Gaussian multiplicative chaos) one seeks to define a measure m h = e gh(z) dz where h is an instance of the Gaussian free field on a planar domain D. Since h is a distribution, not a function, one needs a regularization procedure to make this precise: for example, one may let he(z) be the average value of h on the circle of radius e centered at z (or an analogous average defined using a bump function supported inside that circle) and then write m h = lime!0e g 2 2 e ghe(z) dz. If f : ˜ D! D is a conformal map, one can write ˜ h = h f + Q logjf 0 j, where Q = 2=g +g=2. The measure m ˜ h on ˜ D is then a.s. equivalent to the pullback via f 1 of the measure m h on D. Interestingly, although this a.s. holds for each given f , nobody has ever proved that it a.s. holds simultaneously for all possible f . We will prove that this is indeed the case. This is conceptually important because one frequently defines a quantum surface to be an equivalence class of pairs (D; h) (where pairs such as the (D; h) and ( ˜ D; ˜ h) above are considered equivalent) and it is useful to know that the set of pairs (D;m h ) obtained from the set of pairs (D; h) in an equivalence class is itself an equivalence class with respect to the usual measure pullback relation." @default.
- W2401672900 created "2016-06-24" @default.
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- W2401672900 date "2016-05-19" @default.
- W2401672900 modified "2023-09-27" @default.
- W2401672900 title "Field-measure correspondence in Liouville quantum gravity almost surely commutes with all conformal maps simultaneously" @default.
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