Electronic Journal of Probability

Chordal SLE$_6$ explorations of a quantum disk

Ewain Gwynne and Jason Miller

Full-text: Open access

Abstract

We consider a particular type of $\sqrt{8/3} $-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE$_6$ curve $\eta $ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $\eta ([0,t])$ for each time $t \geq 0$ as well as the law of the left/right $\sqrt{8/3} $-quantum boundary length process for $\eta $.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 66, 24 pp.

Dates
Received: 31 January 2017
Accepted: 22 March 2018
First available in Project Euclid: 26 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1532570594

Digital Object Identifier
doi:10.1214/18-EJP161

Mathematical Reviews number (MathSciNet)
MR3835472

Zentralblatt MATH identifier
06924678

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

Keywords
Liouville quantum gravity quantum disk Schramm-Loewner evolution

Rights
Creative Commons Attribution 4.0 International License.

Citation

Gwynne, Ewain; Miller, Jason. Chordal SLE$_6$ explorations of a quantum disk. Electron. J. Probab. 23 (2018), paper no. 66, 24 pp. doi:10.1214/18-EJP161. https://projecteuclid.org/euclid.ejp/1532570594


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References

  • [1] Juhan Aru, Yichao Huang, and Xin Sun. Two perspectives of the 2D unit area quantum sphere and their equivalence. Comm. Math. Phys., 356(1):261–283, 2017.
  • [2] E. Baur, G. Miermont, and G. Ray. Classification of scaling limits of uniform quadrangulations with a boundary. ArXiv e-prints, August 2016.
  • [3] S. Benoist. Natural parametrization of SLE: the Gaussian free field point of view. ArXiv e-prints, August 2017.
  • [4] N. Berestycki, S. Sheffield, and X. Sun. Equivalence of Liouville measure and Gaussian free field. ArXiv e-prints, October 2014.
  • [5] Jérémie Bettinelli and Grégory Miermont. Compact Brownian surfaces I: Brownian disks. Probab. Theory Related Fields, 167(3–4):555–614, 2017.
  • [6] Nicolas Curien. A glimpse of the conformal structure of random planar maps. Comm. Math. Phys., 333(3):1417–1463, 2015.
  • [7] Nicolas Curien and Jean-François Le Gall. The Brownian plane. J. Theoret. Probab., 27(4):1249–1291, 2014.
  • [8] Julien Dubédat. Duality of Schramm-Loewner evolutions. Ann. Sci. Éc. Norm. Supér. (4), 42(5):697–724, 2009.
  • [9] B. Duplantier, J. Miller, and S. Sheffield. Liouville quantum gravity as a mating of trees. ArXiv e-prints, September 2014.
  • [10] Bertrand Duplantier and Scott Sheffield. Liouville quantum gravity and KPZ. Invent. Math., 185(2):333–393, 2011.
  • [11] E. Gwynne and J. Miller. Convergence of the self-avoiding walk on random quadrangulations to SLE$_{8/3}$ on $\sqrt{8/3} $-Liouville quantum gravity. ArXiv e-prints, August 2016.
  • [12] E. Gwynne and J. Miller. Characterizations of SLE$_{\kappa }$ for $\kappa \in (4,8)$ on Liouville quantum gravity. ArXiv e-prints, January 2017.
  • [13] E. Gwynne and J. Miller. Convergence of percolation on uniform quadrangulations with boundary to SLE$_{6}$ on $\sqrt{8/3} $-Liouville quantum gravity. ArXiv e-prints, January 2017.
  • [14] E. Gwynne and J. Miller. Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk. Annales de l’Institut Henri Poincaré, to appear, 2017.
  • [15] Ewain Gwynne and Jason Miller. Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology. Electron. J. Probab., 22:1–47, 2017.
  • [16] Ewain Gwynne, Nina Holden, Jason Miller, and Xin Sun. Brownian motion correlation in the peanosphere for $\kappa >8$. Ann. Inst. Henri Poincaré Probab. Stat., 53(4):1866–1889, 2017.
  • [17] Y. Huang, R. Rhodes, and V. Vargas. Liouville Quantum Gravity on the unit disk. ArXiv e-prints, February 2015.
  • [18] Jean-Pierre Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Québec, 9(2):105–150, 1985.
  • [19] R. Kenyon, J. Miller, S. Sheffield, and D. B. Wilson. Bipolar orientations on planar maps and SLE$_{12}$. ArXiv e-prints, November 2015.
  • [20] Gregory F. Lawler. Conformally invariant processes in the plane, volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005.
  • [21] Jean-François Le Gall. Uniqueness and universality of the Brownian map. Ann. Probab., 41(4):2880–2960, 2013.
  • [22] Y. Li, X. Sun, and S. S. Watson. Schnyder woods, SLE(16), and Liouville quantum gravity. ArXiv e-prints, May 2017.
  • [23] Grégory Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math., 210(2):319–401, 2013.
  • [24] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric. ArXiv e-prints, July 2015.
  • [25] J. Miller and S. Sheffield. Liouville quantum gravity spheres as matings of finite-diameter trees. ArXiv e-prints, June 2015.
  • [26] J. Miller and S. Sheffield. Imaginary geometry III: reversibility of SLE$_\kappa $ for $\kappa \in (4,8)$. Annals of Mathematics, 184(2):455–486, 2016.
  • [27] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding. ArXiv e-prints, May 2016.
  • [28] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map III: the conformal structure is determined. ArXiv e-prints, August 2016.
  • [29] Jason Miller and Scott Sheffield. Imaginary geometry I: interacting SLEs. Probab. Theory Related Fields, 164(3–4):553–705, 2016.
  • [30] Jason Miller and Scott Sheffield. Imaginary geometry II: Reversibility of $\operatorname{SLE} _{\kappa }(\rho _{1};\rho _{2})$ for $\kappa \in (0,4)$. Ann. Probab., 44(3):1647–1722, 2016.
  • [31] Jason Miller and Scott Sheffield. Quantum Loewner evolution. Duke Math. J., 165(17):3241–3378, 2016.
  • [32] Jason Miller and Scott Sheffield. Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees. Probab. Theory Related Fields, 169(3-4):729–869, 2017.
  • [33] Rémi Rhodes and Vincent Vargas. Gaussian multiplicative chaos and applications: A review. Probab. Surv., 11:315–392, 2014.
  • [34] Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221–288, 2000.
  • [35] Oded Schramm and Scott Sheffield. A contour line of the continuum Gaussian free field. Probab. Theory Related Fields, 157(1–2):47–80, 2013.
  • [36] Scott Sheffield. Gaussian free fields for mathematicians. Probab. Theory Related Fields, 139(3–4):521–541, 2007.
  • [37] Scott Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J., 147(1):79–129, 2009.
  • [38] Scott Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann. Probab., 44(5):3474–3545, 2016.
  • [39] Scott Sheffield. Quantum gravity and inventory accumulation. Ann. Probab., 44(6):3804–3848, 2016.
  • [40] Dapeng Zhan. Duality of chordal SLE. Invent. Math., 174(2):309–353, 2008.
  • [41] Dapeng Zhan. Duality of chordal SLE, II. Ann. Inst. Henri Poincaré Probab. Stat., 46(3):740–759, 2010.