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Geodesics and metric ball boundaries in Liouville quantum gravity

Abstract

Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter \(\gamma \in (0,2)\). We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (Commun Math Phys 378(1):625–689, 2020. arXiv:1909.08588) gives a formula for the a.s. Hausdorff dimension for the boundary of the metric ball stopped when it hits a fixed point in terms of the Hausdorff dimension of the whole LQG surface. We also show that the Hausdorff dimension of the metric ball boundary is carried by points which are not on the boundary of any complementary connected component of the ball.

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Notes

  1. Our axioms for a \(\gamma \)-LQG metric only concern a.s. properties of \(D_h\) when h is a GFF plus a continuous function. So, once we have defined \(D_h\) a.s. when h is a GFF plus a continuous function, we can take D to be any measurable mapping \(\mathcal D'(U) \rightarrow \{\text {continuous metrics on U}\}\) which is a.s. consistent with our given definition when h is a GFF plus a continuous function. In fact, the construction of the metric in [11, 13, 15, 20, 22] only gives an explicit definition of \(D_h\) in the case when h is a GFF plus a continuous function.

  2. We cannot apply Proposition 3.7 instead of Proposition 3.6 here since the radii \(\tau _1\) and \(\tau _2\) are random.

  3. Here and in what follows, for two functions fg of a positive real number x we write \(f(x) = o_x(g(x))\) (resp. \(f(x) = O_x(g(x))\)) if f(x)/g(x) goes to zero (resp. remains bounded) as \(x\rightarrow 0\). The dependencies of the rate of convergence will always be specified unless they are clear from the context.

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Acknowledgements

We thank an anonymous referee for helpful comments on an earlier version of this paper. E.G. was partially supported by a Clay research fellowship and a Trinity college, Cambridge junior research fellowship. J.P. was partially supported by the National Science Foundation under Grant No. 2002159. S.S. was partially supported by NSF Grant DMS 1712862. No code or data was involved in this work.

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Correspondence to Ewain Gwynne.

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Gwynne, E., Pfeffer, J. & Sheffield, S. Geodesics and metric ball boundaries in Liouville quantum gravity. Probab. Theory Relat. Fields 182, 905–954 (2022). https://doi.org/10.1007/s00440-022-01112-5

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Mathematics Subject Classification

  • Primary 60D05 (Geometric probability and stochastic geometry)
  • Secondary 60G60 (Random fields)