Weak LQG metrics and Liouville first passage percolation
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Weak LQG metrics and Liouville first passage percolation Julien Dubédat1 · Hugo Falconet1 · Ewain Gwynne2 Xin Sun1
· Joshua Pfeffer3 ·
Received: 18 July 2019 / Revised: 7 May 2020 © The Author(s) 2020
Abstract For γ ∈ (0, 2), we define a weak γ -Liouville quantum gravity (LQG) metric to be a function h → Dh which takes in an instance of the planar Gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding et al. (Tightness of Liouville first passage percolation for γ ∈ (0, 2), 2019. ArXiv √ e-prints, arXiv:1904.08021). It is also known that these axioms are satisfied for the 8/3-LQG metric constructed by Miller and Sheffield (2013–2016). For any weak γ -LQG metric, we obtain moment bounds for diameters of sets as well as point-to-point, set-to-set, and point-to-set distances. We also show that any such metric is locally bi-Hölder continuous with respect to the Euclidean metric and compute the optimal Hölder exponents in both directions. Finally, we show that LQG geodesics cannot spend a long time near a straight line or the boundary of a metric ball. These results are used in subsequent work by Gwynne and Miller which proves that the weak γ -LQG metric is unique for each γ ∈ (0, 2), which in turn gives the uniqueness of the subsequential limit of Liouville first passage √ percolation. However, most of our results are new even in the special case when γ = 8/3. Keywords Liouville quantum gravity · Gaussian free field · LQG metric · Liouville first passage percolation Mathematics Subject Classification 60D05 (geometric probability) · 60G60 (random fields)
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Ewain Gwynne [email protected]
Extended author information available on the last page of the article
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Weak LQG metrics and subsequential limits of LFPP . . . . 1.3 Quantitative properties of weak LQG metrics . . . . . . . . 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . 2 Subsequential limits of LFPP are weak LQG metrics . . . . . . . 2.1 A localized version of LFPP . . . . . . . . . . . . . . . . . 2.2 Subsequential limits . . . . . . . . . . . . . . . . . . . . . . 2.3 Weyl scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Tightness across scales . . . . . . . . . . . . . . . . . . . . 2.5 Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Measurability . . . . . . . . . . . . . . . . . . . . . . . . . 3 Proofs of quantitative properties of weak LQG metrics . . . . . . 3.1 Estimate for the distance between sets . . . . . . . . . . . . 3.2 Asymptotics of the scaling constants . . . . . . . . . . . . . 3.3 Moment bound for diameters . . . . . . . . . . . . . . . . .
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