Existence and uniqueness of the Liouville quantum gravity metric for γ∈ (0 , 2)


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Authors
Gwynne, E 
Miller, J 
Abstract

jats:titleAbstract</jats:title>jats:pWe show that for each jats:inline-formulajats:alternativesjats:tex-math$$\gamma \in (0,2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo∈</mml:mo> mml:mo(</mml:mo> mml:mn0</mml:mn> mml:mo,</mml:mo> mml:mn2</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, there is a unique metric (i.e., distance function) associated with jats:inline-formulajats:alternativesjats:tex-math$$\gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miγ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) jats:italich</jats:italic>, there is a unique random metric jats:inline-formulajats:alternativesjats:tex-math$$D_h$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:miD</mml:mi> mml:mih</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> associated with the Riemannian metric tensor “jats:inline-formulajats:alternativesjats:tex-math$$e^{\gamma h} (dx^2 + dy^2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msup mml:mie</mml:mi> mml:mrow mml:miγ</mml:mi> mml:mih</mml:mi> </mml:mrow> </mml:msup> mml:mrow mml:mo(</mml:mo> mml:mid</mml:mi> mml:msup mml:mix</mml:mi> mml:mn2</mml:mn> </mml:msup> mml:mo+</mml:mo> mml:mid</mml:mi> mml:msup mml:miy</mml:mi> mml:mn2</mml:mn> </mml:msup> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>” on jats:inline-formulajats:alternativesjats:tex-math$${\mathbb {C}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miC</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> which is characterized by a certain list of axioms: it is locally determined by jats:italich</jats:italic> and it transforms appropriately when either adding a continuous function to jats:italich</jats:italic> or applying a conformal automorphism of jats:inline-formulajats:alternativesjats:tex-math$$\mathbb {C}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miC</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The jats:inline-formulajats:alternativesjats:tex-math$$\gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miγ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-LQG metric can be constructed explicitly as the scaling limit of jats:italicLiouville first passage percolation</jats:italic> (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding et al. (Tightness of Liouville first passage percolation for jats:inline-formulajats:alternativesjats:tex-math$$\gamma \in (0,2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo∈</mml:mo> mml:mo(</mml:mo> mml:mn0</mml:mn> mml:mo,</mml:mo> mml:mn2</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, 2019. <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" ext-link-type="uri" xlink:href="http://arxiv.org/abs/1904.08021">arXiv:1904.08021</jats:ext-link>) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when jats:inline-formulajats:alternativesjats:tex-math$$\gamma = \sqrt{8/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo=</mml:mo> mml:msqrt mml:mrow mml:mn8</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, our metric coincides with the jats:inline-formulajats:alternativesjats:tex-math$$\sqrt{8/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msqrt mml:mrow mml:mn8</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula>-LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general jats:inline-formulajats:alternativesjats:tex-math$$\gamma \in (0,2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo∈</mml:mo> mml:mo(</mml:mo> mml:mn0</mml:mn> mml:mo,</mml:mo> mml:mn2</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, we conjecture that our metric is the Gromov–Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems.</jats:p>

Description
Keywords
4901 Applied Mathematics, 4902 Mathematical Physics, 4904 Pure Mathematics, 49 Mathematical Sciences
Journal Title
Inventiones Mathematicae
Conference Name
Journal ISSN
0020-9910
1432-1297
Volume Title
223
Publisher
Springer Science and Business Media LLC
Sponsorship
European Research Council (804166)