The Annals of Probability

Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces

Ewain Gwynne and Jason Miller

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In a recent series of works, Miller and Sheffield constructed a metric on $\sqrt{8/3}$-Liouville quantum gravity (LQG) under which $\sqrt{8/3}$-LQG surfaces (e.g., the LQG sphere, wedge, cone and disk) are isometric to their Brownian surface counterparts (e.g., the Brownian map, half-plane, plane and disk).

We identify the metric gluings of certain collections of independent $\sqrt{8/3}$-LQG surfaces with boundaries identified together according to LQG length along their boundaries. Our results imply in particular that the metric gluing of two independent instances of the Brownian half-plane along their positive boundaries is isometric to a certain LQG wedge decorated by an independent chordal $\mathrm{SLE}_{8/3}$ curve. If one identifies the two sides of the boundary of a single Brownian half-plane, one obtains a certain LQG cone decorated by an independent whole-plane $\mathrm{SLE}_{8/3}$. If one identifies the entire boundaries of two Brownian half-planes, one obtains a different LQG cone and the interface between them is a two-sided variant of whole-plane $\mathrm{SLE}_{8/3}$.

Combined with another work of the authors, the present work identifies the scaling limit of self-avoiding walk on random quadrangulations with $\mathrm{SLE}_{8/3}$ on $\sqrt{8/3}$-LQG.

Article information

Ann. Probab., Volume 47, Number 4 (2019), 2303-2358.

Received: October 2016
Revised: July 2018
First available in Project Euclid: 4 July 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60J67: Stochastic (Schramm-)Loewner evolution (SLE)

Metric gluing Schramm–Loewner evolution Brownian surfaces Liouville quantum gravity


Gwynne, Ewain; Miller, Jason. Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces. Ann. Probab. 47 (2019), no. 4, 2303--2358. doi:10.1214/18-AOP1309.

Export citation


  • [1] Baur, E., Miermont, G. and Ray, G. (2016). Classification of scaling limits of uniform quadrangulations with a boundary. Available at arXiv:1608.01129.
  • [2] Beffara, V. (2008). The dimension of the SLE curves. Ann. Probab. 36 1421–1452.
  • [3] Bettinelli, J. (2015). Scaling limit of random planar quadrangulations with a boundary. Ann. Inst. Henri Poincaré Probab. Stat. 51 432–477.
  • [4] Bettinelli, J. and Miermont, G. (2017). Compact Brownian surfaces I: Brownian disks. Probab. Theory Related Fields 167 555–614.
  • [5] Borot, G., Bouttier, J. and Guitter, E. (2012). A recursive approach to the $O(n)$ model on random maps via nested loops. J. Phys. A 45 045002.
  • [6] Bouttier, J. and Guitter, E. (2009). Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop. J. Phys. A 42 465208.
  • [7] Caraceni, A. (2015). The geometry of large outerplanar and half-planar maps Ph.D. thesis, Scuola Normale Superiore.
  • [8] Caraceni, A. and Curien, N. (2016). Self-Avoiding Walks on the UIPQ. Availablae at arXiv:1609.00245.
  • [9] Curien, N. and Le Gall, J.-F. (2014). The Brownian plane. J. Theoret. Probab. 27 1249–1291.
  • [10] Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 91–106.
  • [11] Duplantier, B., Miller, J. and Sheffield, S. (2014). Liouville quantum gravity as a mating of trees. Available at arXiv:1409.7055.
  • [12] Duplantier, B. and Sheffield, S. (2011). Liouville quantum gravity and KPZ. Invent. Math. 185 333–393.
  • [13] Gwynne, E., Holden, N. and Sun, X. (2019). A distance exponent for Liouville quantum gravity. Probab. Theory Related Fields 173 931–997.
  • [14] Gwynne, E., Kassel, A., Miller, J. and Wilson, D. B. (2018). Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $\kappa>8$. Comm. Math. Phys. 358 1065–1115.
  • [15] Gwynne, E. and Miller, J. (2016). Convergence of the self-avoiding walk on random quadrangulations to $\mathrm{SLE}_{8/3}$ on $\sqrt{8/3}$-Liouville quantum gravity. Available at arXiv:1608.00956.
  • [16] Gwynne, E. and Miller, J. (2017). Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov–Hausdorff–Prokhorov-uniform topology. Electron. J. Probab. 22 84.
  • [17] Jones, P. W. and Smirnov, S. K. (2000). Removability theorems for Sobolev functions and quasiconformal maps. Ark. Mat. 38 263–279.
  • [18] Kenyon, R., Miller, J., Sheffield, S. and Wilson, D. B. (2019). Bipolar orientations on planar maps and SLE$_{12}$. Ann. Probab. 47 1240–1269.
  • [19] Lawler, G., Schramm, O. and Werner, W. (2003). Conformal restriction: The chordal case. J. Amer. Math. Soc. 16 917–955.
  • [20] Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. Amer. Math. Soc., Providence, RI.
  • [21] Le Gall, J.-F. (2010). Geodesics in large planar maps and in the Brownian map. Acta Math. 205 287–360.
  • [22] Le Gall, J.-F. (2013). Uniqueness and universality of the Brownian map. Ann. Probab. 41 2880–2960.
  • [23] Le Gall, J.-F. (2014). Random geometry on the sphere. In Proceedings of the International Congress of Mathematicians—Seoul 2014 1 421–442. Kyung Moon Sa, Seoul.
  • [24] Le Gall, J.-F. (2017). Brownian disks and the Brownian snake. Available at arXiv:1704.08987.
  • [25] Miermont, G. (2009). Random maps and their scaling limits. In Fractal Geometry and Stochastics IV. Progress in Probability 61 197–224. Birkhäuser, Basel.
  • [26] Miermont, G. (2013). The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 319–401.
  • [27] Miller, J. (2018). Dimension of the SLE light cone, the SLE fan, and $\mathrm{SLE}_{\kappa}(\rho)$ for $\kappa\in(0,4)$ and $\rho\in[{\frac{\kappa}{2}}-4,-2)$. Comm. Math. Phys. 360 1083–1119.
  • [28] Miller, J. and Sheffield, S. (2015). An axiomatic characterization of the Brownian map. Available at arXiv:1506.03806.
  • [29] Miller, J. and Sheffield, S. (2015). Liouville quantum gravity and the Brownian map I: The QLE(8/3, 0) metric. Available at arXiv:1507.00719.
  • [30] Miller, J. and Sheffield, S. (2015). Liouville quantum gravity spheres as matings of finite-diameter trees. Available at arXiv:1506.03804.
  • [31] Miller, J. and Sheffield, S. (2016). Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding. Avaialbe at arXiv:1605.03563.
  • [32] Miller, J. and Sheffield, S. (2016). Avaialbe at arXiv:1608.05391.
  • [33] Miller, J. and Sheffield, S. (2016). Imaginary geometry I: Interacting SLEs. Probab. Theory Related Fields 164 553–705.
  • [34] Miller, J. and Sheffield, S. (2016). Quantum Loewner evolution. Duke Math. J. 165 3241–3378.
  • [35] Miller, J. and Sheffield, S. (2017). Imaginary geometry IV: Interior rays, whole-plane reversibility, and space-filling trees. Probab. Theory Related Fields 169 729–869.
  • [36] Miller, J. and Wu, H. (2017). Intersections of SLE paths: The double and cut point dimension of SLE. Probab. Theory Related Fields 167 45–105.
  • [37] Pommerenke, C. (1992). Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 299. Springer, Berlin.
  • [38] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [39] Rhodes, R. and Vargas, V. (2014). Gaussian multiplicative chaos and applications: A review. Probab. Surv. 11 315–392.
  • [40] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. (2) 161 883–924.
  • [41] Schaeffer, G. (1997). Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees. Electron. J. Combin. 4 20.
  • [42] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288.
  • [43] Schramm, O. and Sheffield, S. (2013). A contour line of the continuum Gaussian free field. Probab. Theory Related Fields 157 47–80.
  • [44] Schramm, O. and Wilson, D. B. (2005). SLE coordinate changes. New York J. Math. 11 659–669.
  • [45] Serlet, L. (1997). A large deviation principle for the Brownian snake. Stochastic Process. Appl. 67 101–115.
  • [46] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521–541.
  • [47] Sheffield, S. (2016). Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann. Probab. 44 3474–3545.
  • [48] Sheffield, S. (2016). Quantum gravity and inventory accumulation. Ann. Probab. 44 3804–3848.
  • [49] Vershynin, R. (2012). Introduction to the non-asymptotic analysis of random matrices. In Compressed Sensing 210–268. Cambridge Univ. Press, Cambridge.