scholarly journals Recursive Self-Similarity for Random Trees, Random Triangulations and Brownian Excursion

1994 ◽  
Vol 22 (2) ◽  
pp. 527-545 ◽  
Author(s):  
David Aldous
Keyword(s):  
Random Trees ◽  
Self Similarity ◽  
Brownian Excursion

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.

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 On the Brownian separable permuton

2019 ◽  
Vol 29 (2) ◽  
pp. 241-266
Author(s):  
Mickaël Maazoun
Keyword(s):  
Hausdorff Dimension ◽  
Probability Measure ◽  
Scaling Limit ◽  
Random Measure ◽  
Local Minima ◽  
Self Similarity ◽  
Brownian Excursion ◽  
Random Probability ◽  
Dimension One ◽  
Totally Disconnected

AbstractThe Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Féray, Gerin and Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated to a Brownian excursion whose strict local minima are decorated with independent and identically distributed signs. As a consequence, its support is almost surely totally disconnected, has Hausdorff dimension one, and enjoys self-similarity properties inherited from those of the Brownian excursion. The density function of the averaged permuton is computed and a connection with the shuffling of the Brownian continuum random tree is explored.

scholarly journals HORTON LAW IN SELF-SIMILAR TREES

Fractals ◽  
2016 ◽  
Vol 24 (02) ◽  
pp. 1650017 ◽  
Author(s):  
YEVGENIY KOVCHEGOV ◽  
ILYA ZALIAPIN
Keyword(s):  
Binary Tree ◽  
Random Trees ◽  
Self Similarity ◽  
Expected Number ◽  
Finite Tree ◽  
Self Similar ◽  
Necessary And Sufficient ◽  
Number Of Branches

Self-similarity of random trees is related to the operation of pruning. Pruning [Formula: see text] cuts the leaves and their parental edges and removes the resulting chains of degree-two nodes from a finite tree. A Horton–Strahler order of a vertex [Formula: see text] and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at [Formula: see text]. A branch is a group of neighboring vertices and edges of the same order. The Horton numbers [Formula: see text] and [Formula: see text] are defined as the expected number of branches of order [Formula: see text], and the expected number of order-[Formula: see text] branches that merged order-[Formula: see text] branches, [Formula: see text], respectively, in a finite tree of order [Formula: see text]. The Tokunaga coefficients are defined as [Formula: see text]. The pruning decreases the orders of tree vertices by unity. A rooted full binary tree is said to be mean-self-similar if its Tokunaga coefficients are invariant with respect to pruning: [Formula: see text]. We show that for self-similar trees, the condition [Formula: see text] is necessary and sufficient for the existence of the strong Horton law: [Formula: see text], as [Formula: see text] for some [Formula: see text] and every [Formula: see text]. This work is a step toward providing rigorous foundations for the Horton law that, being omnipresent in natural branching systems, has escaped so far a formal explanation.

Beckett's Broken Circle: A Literary Fractal

2020 ◽  
Vol 29 (2) ◽  
pp. 196-215
Author(s):  
Luke Connolly
Keyword(s):  
Dynamic Chaos ◽  
Geometric Form ◽  
Self Similarity ◽  
Collision Point ◽  
Stable Order ◽  
Structural Principle ◽  
Recent Interest ◽  
Viable Solution ◽  
Specific Shape

This essay proposes that the picture of a broken circle encountered by Watt during the second part of his tale marks a crucial collision point between Beckett's literary and mathematical interests and triggers a process of fractal scaling self-similarity. Building on recent interest concerning the role of the mathematics and mathematical forms found in Beckett's work, I argue that the broken circle depicted in the picture from Watt is a geometric form which (re)appears within at least three interlocking scales throughout Beckett's novel-length prose: (i) its moment of arrival in the picture from Watt, (ii) a macroscopic reinscription in the names of the protagonists populating the five novels spanning Watt through to The Unnamable and (iii) buried within the narratological depths of How It Is. As a structural principle, the interminable irregularity of fractals offered Beckett a viable solution for what he considered the defining task of the modern artist: ‘to find a form to accommodate the mess’. Moreover, the specific shape selected for his fractal is shown to contain within its geometry one of Beckett's most universal and pressing concerns: the inevitable insufficiency of language. Therefore, although this essay restricts itself to examining Beckett's novel-length prose, the idea of a broken circle fractal promises to provide a valuable heuristic through which to reassess the author's other generic avenues. Fractals thus offer a means through which one can bind together the length and breadth of Beckett's oeuvre without ever reducing dynamic chaos to stable order.

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