Liouville quantum gravity metrics are not doubling. (English) Zbl 07923898
Summary: We observe that non-doubling metric spaces can be characterized as those that contain arbitrarily large sets of approximately equidistant points and use this to show that, for \(\gamma\in(0, 2]\), the \(\gamma\)-Liouville quantum gravity metric is almost surely not doubling and thus cannot be quasisymmetrically embedded into any finite-dimensional Euclidean space. This generalizes the corresponding result of Troscheit [34] for the Brownian map (which is equivalent to the case \(\gamma = \sqrt{8/3}\)).
MSC:
| 60D05 | Geometric probability and stochastic geometry |
| 28A80 | Fractals |
| 83C45 | Quantization of the gravitational field |
References:
| [1] | Elena S. Afanas’eva and Viktoriia V. Bilet, Quasi-symmetric mappings and their generalizations on Riemannian manifolds, J. Math. Sci. (N.Y.) 258 (2021), no. 3, 265-275. MathSciNet: MR4425396 · Zbl 1476.30092 |
| [2] | Patrice Assouad, Pseudodistances, facteurs et dimension métrique, Seminar on Harmonic Analysis (1979-1980) (French), Publ. Math. Orsay 80, vol. 7, Univ. Paris XI, Orsay, 1980, pp. 1-33. MathSciNet: MR613989 · Zbl 0486.54021 |
| [3] | Patrice Assouad, Plongements lipschitziens dans \(\mathbb{R}^n\), Bull. Soc. Math. France 111 (1983), no. 4, 429-448. MathSciNet: MR763553 · Zbl 0597.54015 |
| [4] | Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1964 original. MathSciNet: MR1852066 · Zbl 0984.53001 |
| [5] | Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971, Étude de certaines intégrales singulières. MathSciNet: MR0499948 · Zbl 0224.43006 |
| [6] | Jian Ding, Julien Dubédat, Alexander Dunlap, and Hugo Falconet, Tightness of Liouville first passage percolation for \(\mathit{\gamma} \in(0, 2)\), Publ. Math. Inst. Hautes Études Sci. 132 (2020), 353-403. MathSciNet: MR4179836 · Zbl 1455.82008 |
| [7] | Jian Ding and Ewain Gwynne, The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds, Comm. Math. Phys. 374 (2020), no. 3, 1877-1934. MathSciNet: MR4076090 · Zbl 1436.83024 |
| [8] | Jian Ding and Ewain Gwynne, Uniqueness of the critical and supercritical Liouville quantum gravity metrics, Proc. Lond. Math. Soc. (3) 126 (2023), no. 1, 216-333. MathSciNet: MR4535021 · Zbl 1522.60020 |
| [9] | Jian Ding and Ewain Gwynne, The critical Liouville quantum gravity metric induces the Euclidean topology, Front. Math. 19 (2024), no. 1, 1-46. MathSciNet: MR4688307 · Zbl 1543.60015 |
| [10] | Julien Dubédat, Hugo Falconet, Ewain Gwynne, Joshua Pfeffer, and Xin Sun, Weak LQG metrics and Liouville first passage percolation, Probab. Theory Related Fields 178 (2020), no. 1-2, 369-436. MathSciNet: MR4146541 · Zbl 1469.60054 |
| [11] | Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, and Vincent Vargas, Renormalization of critical Gaussian multiplicative chaos and KPZ relation, Comm. Math. Phys. 330 (2014), no. 1, 283-330. MathSciNet: MR3215583 · Zbl 1297.60033 |
| [12] | Bertrand Duplantier and Scott Sheffield, Liouville quantum gravity and KPZ, Invent. Math. 185 (2011), no. 2, 333-393. MathSciNet: MR2819163 · Zbl 1226.81241 |
| [13] | Jonathan M. Fraser, Assouad dimension and fractal geometry, Cambridge Tracts in Mathematics, vol. 222, Cambridge University Press, Cambridge, 2021. MathSciNet: MR4411274 · Zbl 1467.28001 |
| [14] | Michael Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), no. 2, 179-195. MathSciNet: MR630949 · Zbl 0467.53021 |
| [15] | Ewain Gwynne, Nina Holden, and Xin Sun, Mating of trees for random planar maps and Liouville quantum gravity: a survey, Topics in statistical mechanics (Cédric Boutillier and Kilian Raschel, eds.), Panoramas et Synthèses [Panoramas and Syntheses], vol. 59, Société Mathématique de France, Paris, 2023, pp. 41-120. MathSciNet: MR4619309 · Zbl 1516.82003 |
| [16] | Ewain Gwynne and Jason Miller, Local metrics of the Gaussian free field, Ann. Inst. Fourier (Grenoble) 70 (2020), no. 5, 2049-2075. MathSciNet: MR4245606 · Zbl 1478.60041 |
| [17] | Ewain Gwynne and Jason Miller, Conformal covariance of the Liouville quantum gravity metric for \(\mathit{\gamma} \in(0, 2)\), Ann. Inst. Henri Poincaré Probab. Stat. 57 (2021), no. 2, 1016-1031. MathSciNet: MR4260493 · Zbl 1472.60020 |
| [18] | Ewain Gwynne and Jason Miller, Existence and uniqueness of the Liouville quantum gravity metric for \(\mathit{\gamma} \in(0, 2)\), Invent. Math. 223 (2021), no. 1, 213-333. MathSciNet: MR4199443 · Zbl 1461.83018 |
| [19] | Ewain Gwynne, Jason Miller, and Scott Sheffield, The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity, Ann. Probab. 49 (2021), no. 4, 1677-1717. MathSciNet: MR4260465 · Zbl 1484.60007 |
| [20] | Ewain Gwynne and Joshua Pfeffer, KPZ formulas for the Liouville quantum gravity metric, Trans. Amer. Math. Soc. 375 (2022), no. 12, 8297-8324. MathSciNet: MR4504639 · Zbl 1501.60007 |
| [21] | Nina Holden and Xin Sun, Convergence of uniform triangulations under the Cardy embedding, Acta Math. 230 (2023), no. 1, 93-203. MathSciNet: MR4567714 · Zbl 1514.60016 |
| [22] | Jean-Pierre Kahane, Sur le chaos multiplicatif, Ann. Sci. Math. Québec 9 (1985), no. 2, 105-150. MathSciNet: MR829798 · Zbl 0596.60041 |
| [23] | Jean-François Le Gall, Uniqueness and universality of the Brownian map, Ann. Probab. 41 (2013), no. 4, 2880-2960. MathSciNet: MR3112934 · Zbl 1282.60014 |
| [24] | Jouni Luukkainen and Eero Saksman, Every complete doubling metric space carries a doubling measure, Proc. Amer. Math. Soc. 126 (1998), no. 2, 531-534. MathSciNet: MR1443161 · Zbl 0897.28007 |
| [25] | Grégory Miermont, The Brownian map is the scaling limit of uniform random plane quadrangulations, Acta Math. 210 (2013), no. 2, 319-401. MathSciNet: MR3070569 · Zbl 1278.60124 |
| [26] | Jason Miller and Scott Sheffield, Imaginary geometry I: interacting SLEs, Probab. Theory Related Fields 164 (2016), no. 3-4, 553-705. MathSciNet: MR3477777 · Zbl 1336.60162 |
| [27] | Jason Miller and Scott Sheffield, Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees, Probab. Theory Related Fields 169 (2017), no. 3-4, 729-869. MathSciNet: MR3719057 · Zbl 1378.60108 |
| [28] | Jason Miller and Scott Sheffield, Liouville quantum gravity and the Brownian map I: the \(\operatorname{QLE}(8 / 3, 0) metric \), Invent. Math. 219 (2020), no. 1, 75-152. MathSciNet: MR4050102 · Zbl 1437.83042 |
| [29] | Jason Miller and Scott Sheffield, Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding, Ann. Probab. 49 (2021), no. 6, 2732-2829. MathSciNet: MR4348679 · Zbl 1478.60045 |
| [30] | Jason Miller and Scott Sheffield, Liouville quantum gravity and the Brownian map III: the conformal structure is determined, Probab. Theory Related Fields 179 (2021), no. 3-4, 1183-1211. MathSciNet: MR4242633 · Zbl 1478.60044 |
| [31] | Assaf Naor and Ofer Neiman, Assouad’s theorem with dimension independent of the snowflaking, Rev. Mat. Iberoam. 28 (2012), no. 4, 1123-1142. MathSciNet: MR2990137 · Zbl 1260.46016 |
| [32] | Joshua Pfeffer, Weak Liouville quantum gravity metrics with matter central charge \(\mathbf{c} \in(- \operatorname{\infty}, 25)\), Probab. Math. Phys. 5 (2024), no. 3, 545-608. MathSciNet: MR4765494 · Zbl 07898592 |
| [33] | Scott Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields 139 (2007), no. 3-4, 521-541. MathSciNet: MR2322706 · Zbl 1132.60072 |
| [34] | Sascha Troscheit, On quasisymmetric embeddings of the Brownian map and continuum trees, Probab. Theory Related Fields 179 (2021), no. 3-4, 1023-1046. MathSciNet: MR4242630 · Zbl 1478.60046 |
| [35] | Jussi Väisälä, Quasisymmetric embeddings in Euclidean spaces, Trans. Amer. Math. Soc. 264 (1981), no. 1, 191-204. MathSciNet: MR597876 · Zbl 0456.30018 |
| [36] | A. L. Vol’berg and S. V. Konyagin, On measures with the doubling condition, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 666-675. MathSciNet: MR903629 · Zbl 0649.42010 |
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