Field-measure correspondence in Liouville quantum gravity almost surely commutes with all conformal maps simultaneously

2016/05/19 by Scott Sheffield⋆, Menglu Wang, Sheffield, Scott +1 · 4 citations
Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1605.06171

openalex publication_date 2016/05/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In Liouville quantum gravity (or 2d-Gaussian multiplicative chaos) one seeks to define a measure μ^h = e^γh(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 h_ε(z) be the average value of h on the circle of radius ε centered at z (or an analogous average defined using a bump function supported inside that circle) and then write μ^h = lim_{ε→ 0} ε^(γ2)/(2) e^γh_ε(z) dz. If ϕ: D → D is a conformal map, one can write h = h ∘ ϕ+ Q log |ϕ'|, where Q = 2/γ+ γ/2. The measure μ ^h on D is then a.s. equivalent to the pullback via ϕ^-1 of the measure μ^h on D. Interestingly, although this a.s. holds for each given ϕ, nobody has ever proved that it a.s. holds \textit simultaneously for all possible ϕ. 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,μ^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.

Citations

• Equivalence of Liouville measure and Gaussian free field
• Chaos, concentration, and multiple valleys

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