## 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

Mathematical Reviews number (MathSciNet)
MR3718712

Zentralblatt MATH identifier
1378.60030

#### 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

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