## Electronic Journal of Probability

### Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology

#### Abstract

We prove that the uniform infinite half-plane quadrangulation (UIHPQ), with either general or simple boundary, equipped with its graph distance, its natural area measure, and the curve which traces its boundary, converges in the scaling limit to the Brownian half-plane. The topology of convergence is given by the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) metric on curve-decorated metric measure spaces, which is a generalization of the Gromov-Hausdorff metric whereby two such spaces $(X_1, d_1 , \mu _1,\eta _1)$ and $(X_2, d_2 , \mu _2,\eta _2)$ are close if they can be isometrically embedded into a common metric space in such a way that the spaces $X_1$ and $X_2$ are close in the Hausdorff distance, the measures $\mu _1$ and $\mu _2$ are close in the Prokhorov distance, and the curves $\eta _1$ and $\eta _2$ are close in the uniform distance.

#### Article information

Source
Electron. J. Probab. Volume 22 (2017), paper no. 84, 47 pp.

Dates
Accepted: 2 September 2017
First available in Project Euclid: 14 October 2017

https://projecteuclid.org/euclid.ejp/1507946759

Digital Object Identifier
doi:10.1214/17-EJP102

Zentralblatt MATH identifier
06797894

#### Citation

Gwynne, Ewain; Miller, Jason. Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology. Electron. J. Probab. 22 (2017), paper no. 84, 47 pp. doi:10.1214/17-EJP102. https://projecteuclid.org/euclid.ejp/1507946759

#### References

• [1] Céline Abraham. Rescaled bipartite planar maps converge to the Brownian map.Ann. Inst. Henri Poincaré Probab. Stat., 52(2):575–595, 2016.
• [2] Romain Abraham, Jean-François Delmas, and Patrick Hoscheit. A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces.Electron. J. Probab., 18:no. 14, 21, 2013.
• [3] L. Addario Berry and M. Albenque. The scaling limit of random simple triangulations and random simple quadrangulations.ArXiv e-prints, June 2013.
• [4] E. Baur, G. Miermont, and G. Ray. Classification of scaling limits of uniform quadrangulations with a boundary.ArXiv e-prints, August 2016.
• [5] Itai Benjamini and Oded Schramm. Recurrence of distributional limits of finite planar graphs.Electron. J. Probab., 6:no. 23, 13 pp. (electronic), 2001.
• [6] J. Bertoin and R. A. Doney. On conditioning a random walk to stay nonnegative.Ann. Probab., 22(4):2152–2167, 1994.
• [7] Jérémie Bettinelli. Scaling limits for random quadrangulations of positive genus.Electron. J. Probab., 15:no. 52, 1594–1644, 2010.
• [8] Jérémie Bettinelli. Scaling limit of random planar quadrangulations with a boundary.Ann. Inst. Henri Poincaré Probab. Stat., 51(2):432–477, 2015.
• [9] Jérémie Bettinelli, Emmanuel Jacob, and Grégory Miermont. The scaling limit of uniform random plane maps,viathe Ambjørn-Budd bijection.Electron. J. Probab., 19:no. 74, 16, 2014.
• [10] Jérémie Bettinelli and Grégory Miermont. Compact Brownian surfaces I: Brownian disks.Probab. Theory Related Fields, 167(3-4):555–614, 2017.
• [11] G. Borot, J. Bouttier, and E. Guitter. A recursive approach to the O(n) model on random maps via nested loops.Journal of Physics A Mathematical General, 45(4):045002, February 2012.
• [12] J. Bouttier, P. Di Francesco, and E. Guitter. Planar maps as labeled mobiles.Electron. J. Combin., 11(1):Research Paper 69, 27, 2004.
• [13] J. Bouttier and E. Guitter. Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop.J. Phys. A, 42(46):465208, 44, 2009.
• [14] Dmitri Burago, Yuri Burago, and Sergei Ivanov.A course in metric geometry, volume 33 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001.
• [15] A. Caraceni and N. Curien. Geometry of the Uniform Infinite Half-Planar Quadrangulation.ArXiv e-prints, August 2015.
• [16] A. Caraceni and N. Curien. Self-Avoiding Walks on the UIPQ.ArXiv e-prints, September 2016.
• [17] Alessandra Caraceni.The geometry of large outerplanar and half-planar maps. PhD thesis, Scuola Normale Superiore, 2015.
• [18] Nicolas Curien and Jean-François Le Gall. The Brownian plane.J. Theoret. Probab., 27(4):1249–1291, 2014.
• [19] Nicolas Curien and Grégory Miermont. Uniform infinite planar quadrangulations with a boundary.Random Structures Algorithms, 47(1):30–58, 2015.
• [20] B. Duplantier, J. Miller, and S. Sheffield. Liouville quantum gravity as a mating of trees.ArXiv e-prints, September 2014.
• [21] Bertrand Duplantier and Scott Sheffield. Liouville quantum gravity and KPZ.Invent. Math., 185(2):333–393, 2011.
• [22] B. V. Gnedenko and A. N. Kolmogorov.Limit distributions for sums of independent random variables. Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob.
• [23] Andreas Greven, Peter Pfaffelhuber, and Anita Winter. Convergence in distribution of random metric measure spaces ($\Lambda$-coalescent measure trees).Probab. Theory Related Fields, 145(1-2):285–322, 2009.
• [24] Misha Gromov.Metric structures for Riemannian and non-Riemannian spaces, volume 152 ofProgress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.
• [25] E. Gwynne, N. Holden, and X. Sun. Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense.ArXiv e-prints, March 2016.
• [26] E. Gwynne, N. Holden, and X. Sun. Full scaling limit of site percolation on random triangulations toward SLE$_6$ on $\sqrt{8/3}$-Liouville quantum gravity in the metric and peanosphere sense. In preparation, 2017.
• [27] E. Gwynne, A. Kassel, J. Miller, and D. B. Wilson. Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $\kappa \geq 8$.ArXiv e-prints, March 2016.
• [28] E. Gwynne, C. Mao, and X. Sun. Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times.ArXiv e-prints, February 2015.
• [29] 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.
• [30] E. Gwynne and J. Miller. Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces.ArXiv e-prints, August 2016.
• [31] 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.
• [32] E. Gwynne and J. Miller. Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk.ArXiv e-prints, January 2017.
• [33] E. Gwynne and X. Sun. Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map III: finite volume case.ArXiv e-prints, October 2015.
• [34] E. Gwynne and X. Sun. Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map II: local estimates and empty reduced word exponent.Electronic Jorunal of Probability, 22:Paper No. 45, 1–56, 2017.
• [35] R. Kenyon, J. Miller, S. Sheffield, and D. B. Wilson. Bipolar orientations on planar maps and SLE$_{12}$.ArXiv e-prints, November 2015.
• [36] Jean-François Le Gall. The topological structure of scaling limits of large planar maps.Invent. Math., 169(3):621–670, 2007.
• [37] Jean-François Le Gall. Uniqueness and universality of the Brownian map.Ann. Probab., 41(4):2880–2960, 2013.
• [38] Jean-François Le Gall and Grégory Miermont. Scaling limits of random trees and planar maps. InProbability and statistical physics in two and more dimensions, volume 15 ofClay Math. Proc., pages 155–211. Amer. Math. Soc., Providence, RI, 2012.
• [39] Jiří Matoušek.Lectures on discrete geometry, volume 212 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 2002.
• [40] Grégory Miermont. Tessellations of random maps of arbitrary genus.Ann. Sci. Éc. Norm. Supér. (4), 42(5):725–781, 2009.
• [41] Grégory Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations.Acta Math., 210(2):319–401, 2013.
• [42] J. Miller and S. Sheffield. An axiomatic characterization of the Brownian map.ArXiv e-prints, June 2015.
• [43] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric.ArXiv e-prints, July 2015.
• [44] J. Miller and S. Sheffield. Liouville quantum gravity spheres as matings of finite-diameter trees.ArXiv e-prints, June 2015.
• [45] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding.ArXiv e-prints, May 2016.
• [46] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map III: the conformal structure is determined.ArXiv e-prints, August 2016.
• [47] Jason Miller and Scott Sheffield. Imaginary geometry I: interacting SLEs.Probab. Theory Related Fields, 164(3-4):553–705, 2016.
• [48] Jason Miller and Scott Sheffield. Quantum Loewner evolution.Duke Math. J., 165(17):3241–3378, 2016.
• [49] Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees.Israel J. Math., 118:221–288, 2000.
• [50] Oded Schramm and Scott Sheffield. A contour line of the continuum Gaussian free field.Probab. Theory Related Fields, 157(1-2):47–80, 2013.
• [51] Scott Sheffield. Gaussian free fields for mathematicians.Probab. Theory Related Fields, 139(3-4):521–541, 2007.
• [52] Scott Sheffield. Quantum gravity and inventory accumulation.Ann. Probab., 44(6):3804–3848, 2016.