scholarly journals Almost Giant Clusters for Percolation on Large Trees with Logarithmic Heights

2013 ◽  
Vol 50 (3) ◽  
pp. 603-611 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Trees ◽  
Applied Probability ◽  
Bond Percolation ◽  
General Criterion ◽  
Asymptotic Results ◽  
Percolation Clusters ◽  
Large Trees ◽  
Logarithmic Height

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

scholarly journals Almost Giant Clusters for Percolation on Large Trees with Logarithmic Heights

2013 ◽  
Vol 50 (03) ◽  
pp. 603-611 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Trees ◽  
Applied Probability ◽  
Bond Percolation ◽  
General Criterion ◽  
Asymptotic Results ◽  
Percolation Clusters ◽  
Large Trees ◽  
Logarithmic Height

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

Hydrodynamic radii of diffusion‐limited aggregates and bond‐percolation clusters

1988 ◽  
Vol 89 (9) ◽  
pp. 5887-5889 ◽  
Author(s):  
Zhong‐Ying Chen ◽  
Paul C. Weakliem ◽  
Paul Meakin
Keyword(s):  
Bond Percolation ◽  
Percolation Clusters ◽  
Diffusion Limited

scholarly journals Survival of inhomogeneous Galton-Watson processes

2008 ◽  
Vol 40 (03) ◽  
pp. 798-814
Author(s):  
Erik Broman ◽  
Ronald Meester
Keyword(s):  
Uniform Convergence ◽  
Growth Rates ◽  
Survival Probability ◽  
Critical Value ◽  
Random Trees ◽  
Family Tree ◽  
Percolation Clusters ◽  
The Family ◽  
Watson Process ◽  
The Way

We study the survival properties of inhomogeneous Galton-Watson processes. We determine the so-called branching number (which is the reciprocal of the critical value for percolation) for these random trees (conditioned on being infinite), which turns out to be an almost sure constant. We also shed some light on the way in which the survival probability varies between the generations. When we perform independent percolation on the family tree of an inhomogeneous Galton-Watson process, the result is essentially a family of inhomogeneous Galton-Watson processes, parameterized by the retention probability p. We provide growth rates, uniformly in p, of the percolation clusters, and also show uniform convergence of the survival probability from the nth level along subsequences. These results also establish, as a corollary, the supercritical continuity of the percolation function. Some of our results are generalizations of results in Lyons (1992).

scholarly journals Embedding Small Digraphs and Permutations in Binary Trees and Split Trees

Algorithmica ◽  
2020 ◽  
Vol 82 (3) ◽  
pp. 589-615
Author(s):  
Michael Albert ◽  
Cecilia Holmgren ◽  
Tony Johansson ◽  
Fiona Skerman
Keyword(s):  
Large Class ◽  
Random Permutation ◽  
Random Trees ◽  
Binary Trees ◽  
Independent Interest ◽  
Explicit Formulas ◽  
Random Node ◽  
Labelled Trees ◽  
Logarithmic Height ◽  
Split Trees

AbstractWe investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees (Cai et al. in Combin Probab Comput 28(3):335–364, 2019). We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye (SIAM J Comput 28(2):409–432, 1998. 10.1137/s0097539795283954). Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with probability tending to one as the number of balls increases, the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

The characteristics of den trees used by the squirrel glider (Petaurus norfolcensis) in temperate Australian woodlands

2008 ◽  
Vol 35 (7) ◽  
pp. 663 ◽  
Author(s):  
M. J. Crane ◽  
R. M. Montague-Drake ◽  
R. B. Cunningham ◽  
D. B. Lindenmayer
Keyword(s):  
Wildlife Conservation ◽  
Basal Area ◽  
Multiple Regression Model ◽  
Random Trees ◽  
Eucalyptus Species ◽  
Critical Habitat ◽  
Habitat Features ◽  
Interspecific Differences ◽  
Large Trees ◽  
Squirrel Glider

Being able to recognise critical habitat features such as nesting and denning sites is essential for wildlife conservation. It is particularly true for the den trees of species threatened by habitat loss, such as the squirrel glider (Petaurus norfolcensis). Measurements of 146 den trees of the squirrel glider were made in fragmented woodlands of the South-west Slopes of New South Wales and data compared with random trees to quantify the key characteristics of den sites. The likelihood of use as a den tree increased with increasing numbers of visible tree hollows and tree size. Dieback was also a positive indicator of den use. However, when visible hollows within a tree are abundant, dieback-free trees were preferred. Measures of den tree context such as basal area, the number of neighbouring large trees and distance to the next nearest tree, were also found to be important determinants of the likelihood of usage. The above variables were combined into a multiple regression model. The squirrel glider favoured particular Eucalyptus species and some broader eucalypt groups. We believe such variations were most likely due to interspecific differences in hollow development and dieback among the various groups, rather than bark type, a factor previously cited as an important determinant of den tree usage. The ‘best’ model had high negative predictive power, suggesting it would be useful for identifying (1) trees that could be felled without a loss of this critical habitat resource (e.g. at development sites) and (2) areas unsuitable for potential squirrel glider relocation or habitat enhancement. Squirrel gliders show preference for a combination of tree and tree context features in selecting den trees. Understanding these features will help managers enhance and protect denning resources for this species.

scholarly journals Spreading of Infections on Network Models: Percolation Clusters and Random Trees

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3054
Author(s):  
Hector Eduardo Roman ◽  
Fabrizio Croccolo
Keyword(s):  
Infectious Disease ◽  
Network Models ◽  
Building Blocks ◽  
Sir Model ◽  
Random Trees ◽  
Square Lattice ◽  
Hierarchical Networks ◽  
Tree Structures ◽  
Percolation Clusters ◽  
Suitable Framework

We discuss network models as a general and suitable framework for describing the spreading of an infectious disease within a population. We discuss two types of finite random structures as building blocks of the network, one based on percolation concepts and the second one on random tree structures. We study, as is done for the SIR model, the time evolution of the number of susceptible (S), infected (I) and recovered (R) individuals, in the presence of a spreading infectious disease, by incorporating a healing mechanism for infecteds. In addition, we discuss in detail the implementation of lockdowns and how to simulate them. For percolation clusters, we present numerical results based on site percolation on a square lattice, while for random trees we derive new analytical results, which are illustrated in detail with a few examples. It is argued that such hierarchical networks can complement the well-known SIR model in most circumstances. We illustrate these ideas by revisiting USA COVID-19 data.

scholarly journals A Boltzmann Approach to Percolation on Random Triangulations

2019 ◽  
Vol 71 (1) ◽  
pp. 1-43 ◽  
Author(s):  
Olivier Bernardi ◽  
Nicolas Curien ◽  
Grégory Miermont
Keyword(s):  
Percolation Model ◽  
Bond Percolation ◽  
Critical Percolation ◽  
Site Percolation ◽  
Percolation Clusters ◽  
Percolation Parameter ◽  
Stable Map ◽  
Percolation Thresholds ◽  
Boltzmann Model ◽  
Independent Derivation

AbstractWe study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length$n$decays exponentially with$n$except at a particular value$p_{c}$of the percolation parameter$p$for which the decay is polynomial (of order$n^{-10/3}$). Moreover, the probability that the origin cluster has size$n$decays exponentially if$p<p_{c}$and polynomially if$p\geqslant p_{c}$.The critical percolation value is$p_{c}=1/2$for site percolation, and$p_{c}=(2\sqrt{3}-1)/11$for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds.Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at$p_{c}$, the percolation clusters conditioned to have size$n$should converge toward the stable map of parameter$\frac{7}{6}$introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

scholarly journals Survival of inhomogeneous Galton-Watson processes

2008 ◽  
Vol 40 (3) ◽  
pp. 798-814
Author(s):  
Erik Broman ◽  
Ronald Meester
Keyword(s):  
Uniform Convergence ◽  
Growth Rates ◽  
Survival Probability ◽  
Critical Value ◽  
Random Trees ◽  
Family Tree ◽  
Percolation Clusters ◽  
The Family ◽  
Watson Process ◽  
The Way

We study the survival properties of inhomogeneous Galton-Watson processes. We determine the so-called branching number (which is the reciprocal of the critical value for percolation) for these random trees (conditioned on being infinite), which turns out to be an almost sure constant. We also shed some light on the way in which the survival probability varies between the generations. When we perform independent percolation on the family tree of an inhomogeneous Galton-Watson process, the result is essentially a family of inhomogeneous Galton-Watson processes, parameterized by the retention probability p. We provide growth rates, uniformly in p, of the percolation clusters, and also show uniform convergence of the survival probability from the nth level along subsequences. These results also establish, as a corollary, the supercritical continuity of the percolation function. Some of our results are generalizations of results in Lyons (1992).

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