Geodesic rays in the uniform infinite half-planar
quadrangulation return to the boundary

Baur, Erich; Miermont, Grégory; Richier, Loïc

doi:10.30757/alea.v13-40

Cited by 2 publications

(1 citation statement)

References 15 publications

(52 reference statements)

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“…Remark 6. In [9], it has been proved that geodesics in the standard UIHPQ intersect both the left and right part of the boundary infinitely many times (see [9,Section 2.3.3] for the exact terminology). However, up to removing finite quadrangulations that hang off from the boundary, the UIHPQ has the topology of a half-plane.…”

Section: Tree Structurementioning

confidence: 99%

Uniform infinite half-planar quadrangulations with skewness

Baur¹,

Richier²

2018

Electron. J. Probab.

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We introduce a one-parameter family of random infinite quadrangulations of the half-plane, which we call the uniform infinite half-planar quadrangulations with skewness (UIHPQ p for short, with p P r0, 1{2s measuring the skewness). They interpolate between Kesten's tree corresponding to p " 0 and the usual UIHPQ with a general boundary corresponding to p " 1{2. As we make precise, these models arise as local limits of uniform quadrangulations with a boundary when their volume and perimeter grow in a properly fine-tuned way, and they represent all local limits of (sub)critical Boltzmann quadrangulations whose perimeter tend to infinity. Our main result shows that the family pUIHPQ p q p approximates the Brownian half-planes BHP θ , θ ě 0, recently introduced in [8]. For p ă 1{2, we give a description of the UIHPQ p in terms of a looptree associated to a critical two-type Galton-Watson tree conditioned to survive.

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Section: Tree Structurementioning

confidence: 99%

Uniform infinite half-planar quadrangulations with skewness

Baur¹,

Richier²

2018

Electron. J. Probab.

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Simple peeling of planar maps with application to site percolation

Budd

Curien²

2021

Can. J. Math.-J. Can. Math.

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The peeling process, which describes a step-by-step exploration of a planar map, has been instrumental in addressing percolation problems on random infinite planar maps. Bond and face percolations on maps with faces of arbitrary degree are conveniently studied via so-called lazy-peeling explorations. During such explorations, distinct vertices on the exploration contour may, at latter stage, be identified, making the process less suited to the study of site percolation. To tackle this situation and to explicitly identify site-percolation thresholds, we come back to the alternative “simple” peeling exploration of Angel and uncover deep relations with the lazy-peeling process. Along the way, we define and study the random Boltzmann map of the half-plane with a simple boundary for an arbitrary critical weight sequence. Its construction is nontrivial especially in the “dense regime,” where the half-planar random Boltzmann map does not possess an infinite simple core.

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Geodesic rays in the uniform infinite half-planar quadrangulation return to the boundary

Cited by 2 publications

References 15 publications

Uniform infinite half-planar quadrangulations with skewness

Uniform infinite half-planar quadrangulations with skewness

Simple peeling of planar maps with application to site percolation

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