The spectral dimension of random brushes

Jónsson, Thórdur; Stefánsson, Sigurður Örn

doi:10.1088/1751-8113/41/4/045005

Cited by 8 publications

(12 citation statements)

References 15 publications

(38 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The spectral dimension can take any value greater than one and does not necessarily agree with the Hausdorff dimension. We refer to [6,14,15,22] for discussion of the spectral dimension of several types of random graphs. It would be interesting to calculate the spectral dimension of the trees distributed by ν.…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Markov branching in the vertex splitting model

Stefánsson¹

2012

J. Stat. Mech.

View full text Add to dashboard Cite

We study a special case of the vertex splitting model which is a recent model of randomly growing trees. For any finite maximum vertex degree D, we find a one parameter model, with parameter α ∈ [0, 1] which has a so-called Markov branching property. When D = ∞ we find a two parameter model with an additional parameter γ ∈ [0, 1] which also has this feature. In the case D = 3, the model bears resemblance to Ford's α-model of phylogenetic trees and when D = ∞ it is similar to its generalization, the αγ-model. For α = 0, the model reduces to the well known model of preferential attachment.In the case α > 0, we prove convergence of the finite volume probability measures, generated by the growth rules, to a measure on infinite trees which is concentrated on the set of trees with a single spine. We show that the annealed Hausdorff dimension with respect to the infinite volume measure is 1/α. When γ = 0 the model reduces to a model of growing caterpillar graphs in which case we prove that the Hausdorff dimension is almost surely 1/α and that the spectral dimension is almost surely 2/(1 + α). We comment briefly on the distribution of vertex degrees and correlations between degrees of neighbouring vertices.

show abstract

Section: 2mentioning

confidence: 99%

“…For i ≤ k − 1, n i denotes the total volume of the tree which contains τ i as a subtree and for i ≥ k, n i denotes the total volume of each of the other trees which are attached to the nearest neighbour of the root. 22 term in the sum over ℓ…”

Section: (B2) Thenmentioning

confidence: 99%

Markov branching in the vertex splitting model

Stefánsson¹

2012

J. Stat. Mech.

View full text Add to dashboard Cite

show abstract

“…These definitions need not agree and we will see an example whered s exists and is finite, whereas d s is almost surely infinite. For a discussion of the spectral dimension of some random graph ensembles, see [19,20,26].…”

Section: The Spectral Dimension Of Subcritical Treesmentioning

confidence: 99%

“…The aim is to cut off the branches of the finite outgrowths from the path (r, t) so that only single leaves are left. We then use monotonicity results from [26] to compare return probability generating functions. As before we choose n such that n + 1 < β ≤ n + 2.…”

Section: An Upper Bound Ond Smentioning

confidence: 99%

Condensation in Nongeneric Trees

Jónsson

Stefánsson

2010

J Stat Phys

Self Cite

View full text Add to dashboard Cite

We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical Galton-Watson tree with mean offspring probability m < 1. We calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like (1 − m)N where N is the size of the tree. These trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2β − 2 if the weight w n of a vertex of degree n is asymptotic to n −β .

show abstract

“…In parallel to the above-mentioned progress, mathematical physicists have developed generating function methods to calculate the spectral dimension of random trees, see e.g. [10,11,16,17,24]. The benefits of those methods are their simplicity but the disadvantages are that they do not apply as generally and give somewhat weaker results regarding the existence of d s .…”

Section: Introductionmentioning

confidence: 99%

Random Walk on Random Infinite Looptrees

Björnberg

Stefánsson

2015

J Stat Phys

View full text Add to dashboard Cite

Looptrees have recently arisen in the study of critical percolation on the uniform infinite planar triangulation. Here we consider random infinite looptrees defined as the local limit of the looptree associated with a critical Galton-Watson tree conditioned to be large. We study simple random walk on these infinite looptrees by means of providing estimates on volume and resistance growth. We prove that if the offspring distribution of the Galton-Watson process is in the domain of attraction of a stable distribution with index α ∈ (1, 2] then the spectral dimension of the looptree is 2α/(α + 1).

show abstract

The spectral dimension of random brushes

Abstract: Abstract. We consider a class of random graphs, called random brushes, which are con-

Cited by 8 publications

References 15 publications

Markov branching in the vertex splitting model

Markov branching in the vertex splitting model

Condensation in Nongeneric Trees

Random Walk on Random Infinite Looptrees

Contact Info

Product

Resources

About