Abstract
For gamma in (0,2), we define a weakgamma -Liouville quantum gravity (LQG) metric to be a function hmapsto D_h 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 gamma in (0,2), 2019. ArXiv e-prints, arXiv:1904.08021). It is also known that these axioms are satisfied for the sqrt{8/3}-LQG metric constructed by Miller and Sheffield (2013–2016). For any weak gamma -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 gamma -LQG metric is unique for each gamma in (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 gamma =sqrt{8/3}.
Highlights
Where the integral is interpreted in the sense of distributional pairing
Properties of subsequential limits of Liouville first passage percolation (LFPP) We prove, using a general theorem from [26], that every subsequential limit of LFPP can be realized as a measurable function of the field, so the convergence occurs in probability, not just in distribution
Remark 1.12 (The case when ξ > 2/d2) Throughout this paper, we focus on the case of weak γ -Liouville quantum gravity (LQG) metrics
Summary
Let us discuss the approximations of LQG metrics which we will be interested in. We first need to introduce an exponent which plays a fundamental role in the study of γ -LQG distances. The paper [37] proved several properties of geodesics for any metric associated with γ -LQG which satisfies a similar list of axioms to the ones in our definition of a sγtr=on√g L8Q/3G.3metric; at that point such a metric had only been constructed for It far from obvious that subsequential limits of LFPP satisfy (1.9) The reason for this is that scaling space results in scaling the value of ε in (1.5), which in turn changes the subsequence which we are working with. In the contrast to LFPP, for subsequential limits of LGD the coordinate change relation (1.9) is easy to verify but Weyl scaling (Axiom III) appears to be very difficult to verify, so these subsequential limits are not known to be weak LQG metrics in the sense of this paper. Similar considerations apply to variants of LGD defined using embedded planar maps (such as maps constructed from LQG square subdivision [16,19] or mated-CRT maps [21,27]) instead of Euclidean balls, for these variants tightness has not been checked
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