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The Tutte embedding of the Poisson-Voronoi tessellation of the Brownian disk converges to \(\sqrt{8/3}\)-Liouville quantum gravity. (English) Zbl 1441.60015

A planar map is a graph together with an embedding into the plane so that no two edges cross. Two planar maps are considered to be equivalent if they differ by an orientation preserving homeomorphism of the plane. A planar map can be viewed as a metric measure space by equipping it with the graph distance and assigning each vertex one unit of mass. In recent years, there has been considerable progress in studying the large scale metric behavior of planar maps chosen uniformly at random. Of particular relevance to the present paper are the scaling limit results which give the convergence of uniformly random planar maps towards a continuous object in the Gromov-Hausdorff-Prokhorov topology. The first results of this type are [J.-F. Le Gall, Ann. Probab. 41, No. 4, 2880–2960 (2013; Zbl 1282.60014)] and [G. Miermont, Acta Math. 210, No. 2, 319–401 (2013; Zbl 1278.60124)]. In this case, the limiting object is a random metric measure space with the topology of the sphere, called the Brownian map. The works [Le Gall, loc. cit.] and [Miermont, loc. cit.] have been extended to uniformly random planar maps with several other topologies, including the disk, see [the first two authors, Ann. Inst. Henri Poincaré, Probab. Stat. 55, No. 1, 551–589 (2019; Zbl 1466.60019)], the plane and the half-plane. The limiting objects that one obtains are collectively known as Brownian surfaces. Liouville quantum gravity (LQG) is another theory of random surfaces. To define LQG, one starts with af law of Gaussian free field (GFF) \(h\) on a domain \(\mathcal D\) and then considers the random two-dimensional Riemannian manifold with metric tensor \(\exp\{\gamma\, h(z)\}\,(dx^2+ dy^2)\), where \(\gamma\in (0, 2]\) is a parameter. This definition does not make rigorous mathematical sense since \(h\) is a distribution and not a function. Making rigorous sense of various aspects of LQG has been a major topic of research in recent years. Generally, one writes \(\gamma\)-LQG to refer to LQG surfaces with parameter \(\gamma\). The special case \(\gamma={\sqrt{8/3}}\) has long been known to be special: \({\sqrt{8/3}}\)-LQG surfaces, like Brownian surfaces, and are also called pure LQG surfaces. In fact, in a recent series of works by the second two authors has shown that the Brownian map and the so-called \({\sqrt{8/3}}\)-LQG sphere are in some sense equivalent. The aim of the present paper is to construct the conformal structure of a Brownian surface in an explicit manner. The main result of the paper (Theorem 1.1) states that as \(\lambda\to \infty\) (\(\lambda\) fixed and then pick the Poisson point process with intensity measure given by \(\lambda\) times the area measure on the Brownian surface), the random walk on the adjacency graph of Voronoi cells converges modulo parametrization to a limiting continuous path. One can define the Tutte embedding of the adjacency graph of cells in terms of hitting probabilities for the simple random walk, and Theorem 1.2 states that this Tutte embedding converges to \({\sqrt{8/3}}\)-LQG in an appropriate sense as \(\lambda\to \infty\). The paper is organized as follows. Section 2 gives some notation and recall some facts about metric spaces and \({\sqrt{8/3}}\)-LQG surfaces. In Section 3, the authors prove Theorems 1.1 and 1.2 assuming that a certain moment bound for Voronoi cells is satisfied. In Section 4, the required moment bound is obtained. Section 5 discusses several open problems related to the results of the present paper. Appendix A contains the proofs of several elementary properties of Voronoi cells which follow from basic properties of Brownian surfaces.

MSC:

60D05 Geometric probability and stochastic geometry
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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[1] Aizenman, M.; Burchard, A., Hölder regularity and dimension bounds for random curves, Duke Math. J., 99, 3, 419-453 (1999) · Zbl 0944.60022
[2] Adler, R.J.: An introduction to continuity, extrema, and related topics for general Gaussian processes, volume 12 of Institute of Mathematical Statistics Lecture Notes—Monograph Series. Institute of Mathematical Statistics, Hayward, CA (1990) · Zbl 0747.60039
[3] Adler, Rj; Taylor, Je, Random Fields and Geometry (2007), New York: Springer, New York · Zbl 1149.60003
[4] Ben Arous, G., Fribergh, A.: Biased random walks on random graphs. In: Probability and Statistical Physics in St. Petersburg, volume 91 of Proceedings of Symposia in Pure Mathematics, pp. 99-153. American Mathematical Society, Providence, RI (2016). arXiv:1406.5076 · Zbl 1388.60170
[5] Benjamini, I.; Curien, N., Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points, Geom. Funct. Anal., 23, 2, 501-531 (2013) · Zbl 1274.60143
[6] Berestycki, N., Diffusion in planar Liouville quantum gravity, Ann. Inst. Henri Poincaré Probab. Stat., 51, 3, 947-964 (2015) · Zbl 1325.60125
[7] Biskup, M., Recent progress on the random conductance model, Probab. Surv., 8, 294-373 (2011) · Zbl 1245.60098
[8] Bettinelli, J.; Miermont, G., Compact Brownian surfaces I: Brownian disks, Probab. Theory Relat. Fields, 167, 3-4, 555-614 (2017) · Zbl 1373.60062
[9] Baur, E., Miermont, G., Ray, G.: Classification of scaling limits of uniform quadrangulations with a boundary. ArXiv e-prints (2016). arXiv:1608.01129 · Zbl 1448.60027
[10] Borell, C., The Brunn-Minkowski inequality in Gauss space, Invent. Math., 30, 2, 207-216 (1975) · Zbl 0292.60004
[11] Chapuy, G.: On tessellations of random maps and the \(t_g\)-recurrence. ArXiv e-prints, March (2016). arXiv:1603.07714 · Zbl 1384.05035
[12] Curien, N.; Le Gall, J-F, The Brownian plane, J. Theor. Probab., 27, 4, 1249-1291 (2014) · Zbl 1305.05208
[13] Chassaing, P.; Schaeffer, G., Random planar lattices and integrated superBrownian excursion, Probab. Theory Relat. Fields, 128, 2, 161-212 (2004) · Zbl 1041.60008
[14] Cori, R.; Vauquelin, B., Planar maps are well labeled trees, Can. J. Math., 33, 5, 1023-1042 (1981) · Zbl 0415.05020
[15] Ding, J., Dunlap, A.: Subsequential scaling limits for Liouville graph distance. ArXiv e-prints (2018). arXiv:1812.06921 · Zbl 1466.60204
[16] Ding, J.; Dunlap, A., Liouville first-passage percolation: subsequential scaling limits at high temperature, Ann. Probab., 47, 2, 690-742 (2019) · Zbl 1466.60204
[17] Ding, J., Dubédat, J., Dunlap, A., Falconet, H.: Tightness of Liouville first passage percolation for \(\gamma \in (0,2)\). ArXiv e-prints (2019). arXiv:1904.08021 · Zbl 1455.82008
[18] Dubédat, J., Falconet, H.: Liouville metric of star-scale invariant fields: tails and Weyl scaling. Probab. Theory Relat. Fields (to appear) (2018). arXiv:1809.02607 · Zbl 1434.60284
[19] Dubédat, J., Falconet, H., Gwynne, E., Pfeffer, J., Sun, X.: Weak LQG metrics and Liouville first passage percolation. ArXiv e-prints (2019). arXiv:1905.00380 · Zbl 1469.60054
[20] Ding, J., Goswami, S.: Upper bounds on Liouville first passage percolation and Watabiki’s prediction. Commun. Pure Appl. Math. (to appear) (2016). arXiv:1610.09998 · Zbl 1442.60098
[21] Ding, J., Gwynne, E.: The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds. Commun. Math. Phys. (to appear) (2018). arXiv:1807.01072 · Zbl 1436.83024
[22] Duplantier, B., Miller, J., Sheffield, S.: Liouville quantum gravity as a mating of trees. ArXiv e-prints (2014). arXiv:1409.7055 · Zbl 1503.60003
[23] Duplantier, B.; Sheffield, S., Liouville quantum gravity and KPZ, Invent. Math., 185, 2, 333-393 (2011) · Zbl 1226.81241
[24] Duplantier, B., Random walks and quantum gravity in two dimensions, Phys. Rev. Lett., 81, 25, 5489-5492 (1998) · Zbl 0949.83056
[25] Ding, J., Zeitouni, O., Zhang, F.: Heat kernel for Liouville Brownian motion and Liouville graph distance. Commun. Math. Phys. (to appear) (2018). arXiv:1807.00422 · Zbl 1480.60090
[26] Fernique, X.: Regularité des trajectoires des fonctions aléatoires gaussiennes. Lecture Notes in Mathematics, vol. 480, pp. 1-96 (1975) · Zbl 0331.60025
[27] Gwynne, E., Holden, N., Miller, J.: An almost sure KPZ relation for SLE and Brownian motion. Ann. Probab. (to appear) (2015). arXiv:1512.01223 · Zbl 1455.60111
[28] Gwynne, E., Miller, J.: Convergence of the self-avoiding walk on random quadrangulations to \(\text{SLE}_{8/3}\) on \(\sqrt{8/3} \)-Liouville quantum gravity. ArXiv e-prints (2016). arXiv:1608.00956 · Zbl 1481.60166
[29] Gwynne, E., Miller, J.: Metric gluing of Brownian and \(\sqrt{8/3} \)-Liouville quantum gravity surfaces. Ann. Probab. (to appear) (2016). arXiv:1608.00955 · Zbl 1466.60020
[30] Gwynne, E., Miller, J.: Convergence of percolation on uniform quadrangulations with boundary to \(\text{ SLE }_6\) on \(\sqrt{8/3} \)-Liouville quantum gravity. ArXiv e-prints (2017). arXiv:1701.05175
[31] Gwynne, E.; Miller, J., Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology, Electron. J. Probab., 22, 1-47 (2017) · Zbl 1378.60030
[32] Gwynne, E., Miller, J.: Confluence of geodesics in Liouville quantum gravity for \(\gamma \in (0,2)\). ArXiv e-prints (2019). arXiv:1905.00381 · Zbl 1466.60020
[33] Gwynne, E., Miller, J.: Existence and uniqueness of the Liouville quantum gravity metric for \(\gamma \in (0,2)\). ArXiv e-prints (2019). arXiv:1905.00383 · Zbl 1466.60020
[34] Gwynne, E., Miller, J.: Local metrics of the Gaussian free field. ArXiv e-prints (2019). arXiv:1905.00379 · Zbl 1466.60019
[35] Gwynne, E.; Miller, J., Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk, Ann. Inst. Henri Poincaré Probab. Stat., 55, 1, 551-589 (2019) · Zbl 1466.60019
[36] Gwynne, E., Miller, J., Sheffield, S.: The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity. ArXiv e-prints (2017). arXiv:1705.11161 · Zbl 1484.60007
[37] Gwynne, E., Miller, J., Sheffield, S.: An invariance principle for ergodic scale-free random environments. ArXiv e-prints (2018). arXiv:1807.07515 · Zbl 1500.60059
[38] Gwynne, E.; Miller, J.; Sheffield, S., Harmonic functions on mated-CRT maps, Electron. J. Probab., 24, 58, 55 (2019) · Zbl 1466.60090
[39] Garban, C.; Rhodes, R.; Vargas, V., Liouville Brownian motion, Ann. Probab., 44, 4, 3076-3110 (2016) · Zbl 1393.60015
[40] Guitter, E., On a conjecture by Chapuy about Voronoïcells in large maps, J. Stat. Mech. Theory Exp., 2017, 10, 103401 (2017) · Zbl 1458.05238
[41] Holden, N.; Sun, X., SLE as a mating of trees in Euclidean geometry, Commun. Math. Phys., 364, 1, 171-201 (2018) · Zbl 1408.60073
[42] Kahane, J-P, Sur le chaos multiplicatif, Ann. Sci. Math. Québec, 9, 2, 105-150 (1985) · Zbl 0596.60041
[43] Knizhnik, V.; Polyakov, A.; Zamolodchikov, A., Fractal structure of 2D-quantum gravity, Mod. Phys. Lett A, 3, 8, 819-826 (1988)
[44] Le Gall, J-F, The topological structure of scaling limits of large planar maps, Invent. Math., 169, 3, 621-670 (2007) · Zbl 1132.60013
[45] Le Gall, J-F, Geodesics in large planar maps and in the Brownian map, Acta Math., 205, 2, 287-360 (2010) · Zbl 1214.53036
[46] Le Gall, J-F, Uniqueness and universality of the Brownian map, Ann. Probab., 41, 4, 2880-2960 (2013) · Zbl 1282.60014
[47] Le Gall, J-F, Brownian disks and the Brownian snake, Ann. Inst. Henri Poincaré Probab. Stat., 55, 1, 237-313 (2019) · Zbl 1466.60021
[48] Le Gall, J-F; Paulin, F., Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere, Geom. Funct. Anal., 18, 3, 893-918 (2008) · Zbl 1166.60006
[49] Lawler, Gf; Schramm, O.; Werner, W., Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math., 187, 2, 237-273 (2001) · Zbl 1005.60097
[50] Lawler, Gf; Schramm, O.; Werner, W., Values of Brownian intersection exponents. II. Plane exponents, Acta Math., 187, 2, 275-308 (2001) · Zbl 0993.60083
[51] Lawler, Gf; Schramm, O.; Werner, W., Values of Brownian intersection exponents. III. Two-sided exponents, Ann. Inst. H. Poincaré Probab. Stat., 38, 1, 109-123 (2002) · Zbl 1006.60075
[52] Miermont, G., On the sphericity of scaling limits of random planar quadrangulations, Electron. Commun. Probab., 13, 248-257 (2008) · Zbl 1193.60016
[53] Miermont, G., The Brownian map is the scaling limit of uniform random plane quadrangulations, Acta Math., 210, 2, 319-401 (2013) · Zbl 1278.60124
[54] Marckert, J-F; Mokkadem, A., Limit of normalized quadrangulations: the Brownian map, Ann. Probab., 34, 6, 2144-2202 (2006) · Zbl 1117.60038
[55] Miller, J., Sheffield, S.: An axiomatic characterization of the Brownian map. ArXiv e-prints (2015). arXiv:1506.03806 · Zbl 1478.60043
[56] Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric. Invent. Math. (to appear) (2015). arXiv:1507.00719 · Zbl 1437.83042
[57] Miller, J., Sheffield, S.: Liouville quantum gravity spheres as matings of finite-diameter trees. Ann. Inst. Henri Poincaré Probab. Stat. (to appear) (2015). arXiv:1506.03804 · Zbl 1448.60168
[58] Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding. ArXiv e-prints (2016). arXiv:1605.03563
[59] Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map III: the conformal structure is determined. ArXiv e-prints (2016). arXiv:1608.05391 · Zbl 1478.60044
[60] Miller, J.; Sheffield, S., Imaginary geometry I: interacting SLEs, Probab. Theory Relat. Fields, 164, 3-4, 553-705 (2016) · Zbl 1336.60162
[61] Miller, J.; Sheffield, S., Quantum Loewner evolution, Duke Math. J., 165, 17, 3241-3378 (2016) · Zbl 1364.82023
[62] Mullin, Rc, On the enumeration of tree-rooted maps, Can. J. Math., 19, 174-183 (1967) · Zbl 0148.17705
[63] Polyakov, Am, Quantum geometry of bosonic strings, Phys. Lett. B, 103, 3, 207-210 (1981)
[64] Polyakov, Am, Quantum geometry of fermionic strings, Phys. Lett. B, 103, 3, 211-213 (1981)
[65] Rhodes, R.; Vargas, V., Gaussian multiplicative chaos and applications: a review, Probab. Surv., 11, 315-392 (2014) · Zbl 1316.60073
[66] Schaeffer, G.: Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees. Electron. J. Combin. 4(1), Research Paper 20 (electronic) (1997) · Zbl 0885.05076
[67] Sudakov, V.N., Cirel’ son, B.S.: Extremal properties of half-spaces for spherically invariant measures. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 41, 14-24, 165 (1974). (Problems in the theory of probability distributions, II) · Zbl 0351.28015
[68] Sheffield, S., Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab., 44, 5, 3474-3545 (2016) · Zbl 1388.60144
[69] Tutte, Wt, On the enumeration of planar maps, Bull. Am. Math. Soc., 74, 64-74 (1968) · Zbl 0157.31101
[70] Yadin, A.; Yehudayoff, A., Loop-erased random walk and Poisson kernel on planar graphs, Ann. Probab., 39, 4, 1243-1285 (2011) · Zbl 1234.60036
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