## Electronic Journal of Probability

### The UIPQ seen from a point at infinity along its geodesic ray

Daphné Dieuleveut

#### Abstract

We consider the uniform infinite quadrangulation of the plane (UIPQ). Curien, Ménard and Miermont recently established that in the UIPQ, all infinite geodesic rays originating from the root are essentially similar, in the sense that they have an infinite number of common vertices. In this work, we identify the limit quadrangulation obtained by rerooting the UIPQ at a point at infinity on one of these geodesics. More precisely, calling $v_k$ the $k$-th vertex on the “leftmost” geodesic ray originating from the root, and $Q_{\infty }^{(k)}$ the UIPQ re-rooted at $v_k$, we study the local limit of $Q_{\infty }^{(k)}$. To do this, we split the UIPQ along the geodesic ray $(v_k)_{k\geq 0}$. Using natural extensions of the Schaeffer correspondence with discrete trees, we study the quadrangulations obtained on each “side” of this geodesic ray. We finally show that the local limit of $Q_{\infty }^{(k)}$ is the quadrangulation obtained by gluing the limit quadrangulations back together.

#### Article information

Source
Electron. J. Probab. Volume 21, Number (2016), paper no. 54, 44 pp.

Dates
Accepted: 25 July 2016
First available in Project Euclid: 6 September 2016

http://projecteuclid.org/euclid.ejp/1473188081

Digital Object Identifier
doi:10.1214/16-EJP4730

Subjects
Primary: 60C05: Combinatorial probability

#### Citation

Dieuleveut, Daphné. The UIPQ seen from a point at infinity along its geodesic ray. Electron. J. Probab. 21 (2016), paper no. 54, 44 pp. doi:10.1214/16-EJP4730. http://projecteuclid.org/euclid.ejp/1473188081.

#### References

• [1] Omer Angel, Growth and percolation on the uniform infinite planar triangulation, Geometry & Functional Analysis 13 (2003), no. 5, 935–974.
• [2] Omer Angel, Scaling of percolation on infinite planar maps, (2005), arXiv:math/0501006
• [3] Omer Angel and Oded Schramm, Uniform infinite planar triangulations, Communications in Mathematical Physics 241 (2003), no. 2-3, 191–213.
• [4] Philippe Chassaing and Bergfinnur Durhuus, Local limit of labeled trees and expected volume growth in a random quadrangulation, The Annals of Probability 34 (2006), no. 3, 879–917.
• [5] Philippe Chassaing and Gilles Schaeffer, Random planar lattices and integrated superbrownian excursion, Probability Theory and Related Fields (2004), no. 128, 161–212.
• [6] Robert Cori and Bernard Vauquelin, Planar maps are well labeled trees, Canadian Journal of Mathematics 33 (1981), no. 5, 1023–1042.
• [7] Nicolas Curien and Jean-François Le Gall, The Brownian plane, Journal of Theoretical Probability 27 (2014), no. 4, 1249–1291.
• [8] Nicolas Curien and Grégory Miermont, Uniform infinite planar quadrangulations with a boundary, Random Structures and Algorithms 47 (2015), no. 1, 30–58.
• [9] Nicolas Curien, Laurent Ménard, and Grégory Miermont, A view from infinity of the infinite planar quadrangulation, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 45–88.
• [10] Harry Kesten, Subdiffusive behavior of random walk on a random cluster, Annales de l’I.H.P., section B 22 (1986), no. 4, 425–487.
• [11] Maxim Krikun, Local structure of random quadrangulations, (2006). arXiv:math/0512304
• [12] John Lamperti, A new class of probability limit theorems, Bulletin of the American Mathematical Society 67 (1961), no. 3, 267–269.
• [13] Laurent Ménard, The two uniform infinite quadrangulations of the plane have the same law, Annales de l’I.H.P - Probabilités et Statistiques 46 (2010), no. 1, 190–208.
• [14] Jim Pitman, Combinatorial stochastic processes, Springer-Verlag, 2006.
• [15] Gilles Schaeffer, Conjugaison d’arbres et cartes planaires aléatoires, (1998), P.h.D. thesis.