Spectral Dimension of Trees with a Unique Infinite Spine
Using generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically distributed, then both the Hausdorff and spectral dimension...
| Príomhchruthaitheoirí: | , |
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| Formáid: | Journal article |
| Teanga: | English |
| Foilsithe / Cruthaithe: |
2012
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| _version_ | 1826290684075704320 |
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| author | Stefánsson, S Zohren, S |
| author_facet | Stefánsson, S Zohren, S |
| author_sort | Stefánsson, S |
| collection | OXFORD |
| description | Using generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically distributed, then both the Hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex, provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n. We apply this new method to both critical non-generic trees and the attachment and grafting model, which is a special case of the vertex splitting model, resulting in a simplified proof for the values of the Hausdorff and spectral dimension for the former and novel results for the latter. © 2012 Springer Science+Business Media, LLC. |
| first_indexed | 2024-03-07T02:47:57Z |
| format | Journal article |
| id | oxford-uuid:acaa1d04-785d-4833-a7df-dfca5c72a12b |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T02:47:57Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:acaa1d04-785d-4833-a7df-dfca5c72a12b2022-03-27T03:30:33ZSpectral Dimension of Trees with a Unique Infinite SpineJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:acaa1d04-785d-4833-a7df-dfca5c72a12bEnglishSymplectic Elements at Oxford2012Stefánsson, SZohren, SUsing generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically distributed, then both the Hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex, provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n. We apply this new method to both critical non-generic trees and the attachment and grafting model, which is a special case of the vertex splitting model, resulting in a simplified proof for the values of the Hausdorff and spectral dimension for the former and novel results for the latter. © 2012 Springer Science+Business Media, LLC. |
| spellingShingle | Stefánsson, S Zohren, S Spectral Dimension of Trees with a Unique Infinite Spine |
| title | Spectral Dimension of Trees with a Unique Infinite Spine |
| title_full | Spectral Dimension of Trees with a Unique Infinite Spine |
| title_fullStr | Spectral Dimension of Trees with a Unique Infinite Spine |
| title_full_unstemmed | Spectral Dimension of Trees with a Unique Infinite Spine |
| title_short | Spectral Dimension of Trees with a Unique Infinite Spine |
| title_sort | spectral dimension of trees with a unique infinite spine |
| work_keys_str_mv | AT stefanssons spectraldimensionoftreeswithauniqueinfinitespine AT zohrens spectraldimensionoftreeswithauniqueinfinitespine |