scholarly journals The distance profile of rooted and unrooted simply generated trees

2021 ◽  
pp. 1-43
Author(s):  
Gabriel Berzunza Ojeda ◽  
Svante Janson
Keyword(s):  
Local Time ◽  
Random Function ◽  
Convergence Result ◽  
Rooted Trees ◽  
Height Profile ◽  
Hölder Continuous ◽  
Brownian Excursion ◽  
Open Questions ◽  
A Minor ◽  
Simply Generated Trees

Abstract It is well known that the height profile of a critical conditioned Galton–Watson tree with finite offspring variance converges, after a suitable normalisation, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of any order $\alpha<1$ , and that it is a.e. differentiable. We note that it cannot be differentiable at 0, but leave as open questions whether it is Lipschitz, and whether it is continuously differentiable on the half-line $(0,\infty)$ . The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest.

scholarly journals The profile of unlabeled trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Bernhard Gittenberger
Keyword(s):  
Local Time ◽  
Joint Distribution ◽  
Random Trees ◽  
Rooted Trees ◽  
Brownian Excursion ◽  
Level Number ◽  
International Audience ◽  
The Times ◽  
Simply Generated Trees

International audience We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by $\sqrt{n}$) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for simply generated trees and forests to the case of unlabeled rooted trees.

Limiting diffusion for random walks with drift conditioned to stay positive

1978 ◽  
Vol 15 (02) ◽  
pp. 280-291 ◽  
Author(s):  
Peichuen Kao
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Function ◽  
Random Variables ◽  
Principal Result ◽  
Hitting Time ◽  
Brownian Excursion ◽  
Norming Constant ◽  
Finite Dimensional ◽  
Independent Identically Distributed

Let {ξ k : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ 1} = μ ≠ 0. Form the random walk {S n : n ≧ 0} by setting S 0, S n = ξ 1 + ξ 2 + ··· + ξ n , n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ 1) that the finite-dimensional distributions of Xn , conditioned on n &lt; N &lt; ∞ converge to those of the Brownian excursion process.

The Brownian excursion multi-dimensional local time density

1999 ◽  
Vol 36 (2) ◽  
pp. 350-373 ◽  
Author(s):  
Bernhard Gittenberger ◽  
Guy Louchard
Keyword(s):  
Limit Theorem ◽  
Local Time ◽  
Direct Method ◽  
Brownian Excursion ◽  
A Direct Method ◽  
Time Density

Expressions for the multi-dimensional densities of Brownian excursion local time are derived by two different methods: a direct method based on Kac's formula for Brownian functionals and an indirect one based on a limit theorem for Galton–Watson trees.

scholarly journals Mineralogical and geochemical aspects of impact craters

2002 ◽  
Vol 66 (5) ◽  
pp. 745-768 ◽  
Author(s):  
C. Koeberl
Keyword(s):  
Isotopic Method ◽  
Geological Processes ◽  
Open Questions ◽  
Transmission Electron ◽  
Deformation Features ◽  
X Ray Computed ◽  
Imaging And Spectroscopy ◽  
A Minor ◽  
Cathodoluminescence Imaging ◽  
The Impact

AbstractThe importance of impact cratering on terrestrial planets is obvious from the abundance of craters on their surfaces. On Earth, active geological processes rapidly obliterate the cratering record. To date only about 170 impact structures have been recognized on the Earth's surface. Mineralogical, petrographic, and geochemical criteria are used to identify the impact origin of such structures or related ejecta layers. The two most important criteria are the presence of shock metamorphic effects in mineral and rock inclusions in breccias and melt rocks, as well as the demonstration, by geochemical techniques, that these rocks contain a minor extraterrestrial component. There is a variety of macroscopic and microscopic shock metamorphic effects. The most important ones include the presence of planar deformation features in rock-forming minerals, high-pressure polymorphs (e.g. of coesite and stishovite from quartz, or diamond from graphite), diaplectic glass, and rock and mineral melts. These features have been studied by traditional methods involving the petrographic microscope, and more recently with a variety of instrumental techniques, including transmission electron microscopy, Raman spectroscopy, cathodoluminescence imaging and spectroscopy, and high-resolution X-ray computed tomography. Geochemical methods to detect an extraterrestrial component include measurements of the concentrations of siderophile elements, mainly of the platinum-group elements (PGEs), and, more recently, chromium and osmium isotopic studies. The latter two methods can provide confirmation that these elements are actually of meteoritic origin. The Cr isotopic method is also capable of providing information on the meteorite type. In impact studies there is now a trend towards the use of interdisciplinary and multi-technique approaches to solve open questions.

scholarly journals On the distribution of the number of vertices in layers of random trees

1991 ◽  
Vol 4 (3) ◽  
pp. 175-186 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Asymptotic Behavior ◽  
Branching Processes ◽  
Random Trees ◽  
Rooted Trees ◽  
Brownian Excursion

Denote by Sn the set of all distinct rooted trees with n labeled vertices. A tree is chosen at random in the set Sn, assuming that all the possible nn−1 choices are equally probable. Define τn(m) as the number of vertices in layer m, that is, the number of vertices at a distance m from the root of the tree. The distance of a vertex from the root is the number of edges in the path from the vertex to the root. This paper is concerned with the distribution and the moments of τn(m) and their asymptotic behavior in the case where m=[2αn], 0<α<∞ and n→∞. In addition, more random trees, branching processes, the Bernoulli excursion and the Brownian excursion are also considered.

scholarly journals Pattern distribution in various types of random trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Gerard Kok
Keyword(s):  
Normal Distribution ◽  
Limiting Distribution ◽  
Random Trees ◽  
Rooted Trees ◽  
Finite Tree ◽  
International Audience ◽  
Pattern Distribution ◽  
Simply Generated Trees

International audience Let $\mathcal{T}_n$ denote the set of unrooted unlabeled trees of size $n$ and let $\mathcal{M}$ be a particular (finite) tree. Assuming that every tree of $\mathcal{T}_n$ is equally likely, it is shown that the number of occurrences $X_n$ of $\mathcal{M}$ as an induced sub-tree satisfies $\mathbf{E} X_n \sim \mu n$ and $\mathbf{V}ar X_n \sim \sigma^2 n$ for some (computable) constants $\mu > 0$ and $\sigma \geq 0$. Furthermore, if $\sigma > 0$ then $(X_n - \mathbf{E} X_n) / \sqrt{\mathbf{V}ar X_n}$ converges to a limiting distribution with density $(A+Bt^2)e^{-Ct^2}$ for some constants $A,B,C$. However, in all cases in which we were able to calculate these constants, we obtained $B=0$ and thus a normal distribution. Further, if we consider planted or rooted trees instead of $T_n$ then the limiting distribution is always normal. Similar results can be proved for planar, labeled and simply generated trees.

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