## The Annals of Probability

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

#### Abstract

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

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

Dates
Revised: July 2018
First available in Project Euclid: 4 July 2019

https://projecteuclid.org/euclid.aop/1562205710

Digital Object Identifier
doi:10.1214/18-AOP1309

Mathematical Reviews number (MathSciNet)
MR3980922

Zentralblatt MATH identifier
07114718

#### Citation

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. https://projecteuclid.org/euclid.aop/1562205710

#### References

• [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.