Geometry of the Uniform Infinite Half-Planar Quadrangulation

Caraceni, Alessandra; Curien, Nicolas

doi:10.48550/arxiv.1508.00133

Cited by 5 publications

(23 citation statements)

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“…A new construction of the UIHPQ which is better suited to study the metric balls around the root has recently been given in [7]. Although we will work with the first construction, we adopt some notation from there.…”

Section: Planar Maps and Quadrangulations With A Boundarymentioning

confidence: 99%

“…As in [10] or [7], we will work with so-called treed bridges. We will only need their infinite versions, which we define next.…”

Section: A Schaeffer-type Construction 221 Well-labeled Trees and Inf...mentioning

confidence: 99%

“…We stress that usually (e.g., in [10], or in [7]), the Bouttier-Di Francesco-Guitter mapping is first introduced as a bijection between finite versions of treed bridges and (rooted and pointed) finite-size quadrangulations with a boundary. Then it is argued that the mapping can be extended to elements in TB −∞ , yielding infinite quadrangulations.…”

Section: The Bouttier-di Francesco-guitter Mappingmentioning

confidence: 99%

“…The vertex set of such a representation of (b, T ) is therefore given by Z and the union of the tree vertices of T (i), i ∈ DS(b), where we interpret the root of T (i) and the vertex i ∈ Z as one and the same vertex. Following the wording of [7], we call the vertices which belong to the trees T (i), i ∈ DS(b), real vertices, and the vertices j ∈ Z above which no trees are grafted, i.e., the vertices j that do not correspond to down-steps of b, phantom vertices. A corner of (the representation of) (b, T ) is an angular sector between two consecutive edges, in the clockwise contour or left-to-right order.…”

Section: The Bouttier-di Francesco-guitter Mappingmentioning

confidence: 99%

“…In a recent work [7], Caraceni and Curien showed that the analog coalescence property of geodesics holds in the half-planar model UIHPQ: There is with probability one an infinite sequence of distinct vertices, which are all hit by every geodesic ray emanating from the root. Our main result of this paper shows that this property holds in the UIHPQ in a very strong sense.…”

Section: Introductionmentioning

confidence: 99%

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

Baur¹,

Miermont²,

Richier³

2016

ALEA

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We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely distributed along the boundary. As an intermediate step, we show that geodesic rays in the UIHPQ are proper, a fact that was recently established by Caraceni and Curien in [7] by a reasoning different from ours. Finally, we argue that geodesic rays in the uniform infinite half-planar triangulation behave in a very similar manner, even in a strong quantitative sense.

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Section: Planar Maps and Quadrangulations With A Boundarymentioning

confidence: 99%

“…As in [10] or [7], we will work with so-called treed bridges. We will only need their infinite versions, which we define next.…”

Section: A Schaeffer-type Construction 221 Well-labeled Trees and Inf...mentioning

confidence: 99%

Section: The Bouttier-di Francesco-guitter Mappingmentioning

confidence: 99%

Section: The Bouttier-di Francesco-guitter Mappingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Baur¹,

Miermont²,

Richier³

2016

ALEA

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Self-Avoiding Walks on the UIPQ

Caraceni¹,

Curien²

2019

Springer Proceedings in Mathematics &Amp; Statistics

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We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two-sided self-avoiding walk (SAW), which can also be described as the result of glueing together two independent uniform infinite quadrangulations of the half-plane (UIHPQs). We prove a lower bound on the displacement of the SAW which, combined with the estimates of [15], shows that the self-avoiding walk is diffusive. As a byproduct this implies that the volume growth exponent of the lattice in question is 4 (as is the case for the standard UIPQ); nevertheless, using our previous work [9] we show its law to be singular with respect to that of the standard UIPQ, that is -in the language of statistical physicsthe fact that disorder holds. arXiv:1609.00245v2 [math.PR]

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Classification of scaling limits of uniform quadrangulations with a boundary

2019

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We study non-compact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the infinite continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter θ and the infinite-volume Brownian disk of perimeter σ. We also obtain various coupling and limit results clarifying the relation between these objects. (GRAAL) and ANR-15-CE40-0013 (Liouville), and of Fondation Simone et Cino Del Duca.Definition 2.1. Let T > 0. The continuum random tree CRT T with volume T is the random rooted real tree (T X , d x , ρ) for the probability distribution that makes the canonical process X of C([0, T ], R) the standard Brownian excursion with duration T .The term "CRT" usually denotes CRT 1 with volume T = 1, in which case X is taken under the law of the normalized Brownian excursion. We simply write CRT instead of CRT 1 .Note the scaling relation, for λ, T > 0:This comes from the fact that, if e T is a Brownian excursion with duration T , then λe T (·/λ 2 ) has same distribution as e λ 2 T .We should also discuss the role of ρ in the above definition. The re-rooting property of CRT T [2, (20)] states, roughly speaking, that if ρ is a random variable with distribution µ X /µ X (1) (the normalized version of the measure defined above), then (T X , d X , ρ ) has same distribution as (T X , d X , ρ). In this sense, the point ρ plays no distinguished role in the construction of CRT T .Brownian map BM T , T > 0. The Brownian map is roughly speaking the metric gluing of the tree coded by a snake driven by a normalized Brownian excursion, on the tree coded by the excursion itself.Definition 2.2. The Brownian map BM T with volume T is the metric space (M X,W , D X,W , ρ) for the probability law that makes X a Brownian excursion of duration T , and W is the snake driven by X.We write BM instead of BM 1 . The scaling properties of Gaussian processes imply easily that for λ > 0, λ · BM T = d BM λ 4 T .Just as for CRT T , the point ρ in BM T should be seen as a random choice according to the normalized measure µ X,W /µ X,W (1), which is known as the re-rooting property of the Brownian map (Theorem 8.1 of [28]). The latter is a crucial property for characterizing the Brownian map, see, e.g., the recent work [35].Brownian disk BD T,σ , σ ∈ (0, ∞), T > 0. The Brownian disk first appears in [7] as limiting metric space along suitable infinite subsequences. Uniqueness of the limit and a concrete description of the metric were obtained in [9].The description is slightly more elaborate than that of the Brownian map. For t ≥ 0, we let X t = inf [0,t] X.Definition 2.3. The Brownian disk BD T,σ with volume T and boundary length σ is the metric space (M X,W , D X,W , ρ) under the probability measures that makes X a f...

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Geometry of the Uniform Infinite Half-Planar Quadrangulation

Cited by 5 publications

References 0 publications

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

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

Self-Avoiding Walks on the UIPQ

Classification of scaling limits of uniform quadrangulations with a boundary

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