| 1. |
- Björnberg, Jakob, 1983, et al.
(författare)
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STABLE SHREDDED SPHERES AND CAUSAL RANDOM MAPS WITH LARGE FACES
- 2022
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Ingår i: Annals of Probability. - 2168-894X .- 0091-1798. ; 50:5, s. 2056-2084
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a new familiy of random compact metric spaces Sα for α ∈ (1, 2), which we call stable shredded spheres. They are constructed from excursions of α-stable Lévy processes on [0, 1] possessing no negative jumps. Informally, viewing the graph of the Lévy excursion in the plane, each jump of the process is “cut open” and replaced by a circle, and then all points on the graph at equal height, which are not separated by a jump, are identified. We show that the shredded spheres arise as scaling limits of models of causal random planar maps with large faces introduced by Di Francesco and Guitter. We also establish that their Hausdorff dimension is almost surely equal to α. Point identification in the shredded spheres is intimately connected to the presence of decrease points in stable spectrally positive Lévy processes, as studied by Bertoin in the 1990s.
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| 2. |
- Janson, Svante, et al.
(författare)
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Random trees with superexponential branching weights
- 2011
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Ingår i: Journal of Physics A. - : IOP Publishing. - 1751-8113 .- 1751-8121. ; 44:48, s. 485002-
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Tidskriftsartikel (refereegranskat)abstract
- We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors w(n) associated with the vertices of the tree and depending only on their individual degrees n. We focus on the case when w(n) grows faster than exponentially with n. In this case, the measures on trees of finite size N converge weakly as N tends to infinity to a measure which is concentrated on a single tree with one vertex of infinite degree. For explicit weight factors of the form w(n) = (( n - 1)!)(alpha) with alpha > 0, we obtain more refined results about the approach to the infinite volume limit.
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| 3. |
- Janson, Svante, et al.
(författare)
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Scaling limits of random planar maps with a unique large face
- 2015
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Ingår i: Annals of Probability. - 0091-1798 .- 2168-894X. ; 43:3, s. 1045-1081
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Tidskriftsartikel (refereegranskat)abstract
- We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n(-1/2) is described by a Brownian excursion. The planar maps, with the graph metric resealed by n(-1/2), are then shown to converge in distribution toward Aldous' Brownian tree in the Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di Francesco-Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.
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