Random recursive trees and the Bolthausen-Sznitman coalescent
We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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2005
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| _version_ | 1826287804961783808 |
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| author | Goldschmidt, C Martin, J |
| author_facet | Goldschmidt, C Martin, J |
| author_sort | Goldschmidt, C |
| collection | OXFORD |
| description | We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the final collision converges as n tends to infinity, and obtain a scaling law for the sizes of these blocks. We also consider the discrete-time Markov chain giving the number of blocks after each collision of the coalescent restricted to [n]; we show that the transition probabilities of the time-reversal of this Markov chain have limits as n tends to infinity. These results can be interpreted as describing a ``post-gelation'' phase of the Bolthausen-Sznitman coalescent, in which a giant cluster containing almost all of the mass has already formed and the remaining small blocks are being absorbed. |
| first_indexed | 2024-03-07T02:04:09Z |
| format | Journal article |
| id | oxford-uuid:9e630823-e267-4086-9e12-a5eb7ca25dbf |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T02:04:09Z |
| publishDate | 2005 |
| record_format | dspace |
| spelling | oxford-uuid:9e630823-e267-4086-9e12-a5eb7ca25dbf2022-03-27T00:49:45ZRandom recursive trees and the Bolthausen-Sznitman coalescentJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:9e630823-e267-4086-9e12-a5eb7ca25dbfEnglishSymplectic Elements at Oxford2005Goldschmidt, CMartin, JWe describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the final collision converges as n tends to infinity, and obtain a scaling law for the sizes of these blocks. We also consider the discrete-time Markov chain giving the number of blocks after each collision of the coalescent restricted to [n]; we show that the transition probabilities of the time-reversal of this Markov chain have limits as n tends to infinity. These results can be interpreted as describing a ``post-gelation'' phase of the Bolthausen-Sznitman coalescent, in which a giant cluster containing almost all of the mass has already formed and the remaining small blocks are being absorbed. |
| spellingShingle | Goldschmidt, C Martin, J Random recursive trees and the Bolthausen-Sznitman coalescent |
| title | Random recursive trees and the Bolthausen-Sznitman coalescent |
| title_full | Random recursive trees and the Bolthausen-Sznitman coalescent |
| title_fullStr | Random recursive trees and the Bolthausen-Sznitman coalescent |
| title_full_unstemmed | Random recursive trees and the Bolthausen-Sznitman coalescent |
| title_short | Random recursive trees and the Bolthausen-Sznitman coalescent |
| title_sort | random recursive trees and the bolthausen sznitman coalescent |
| work_keys_str_mv | AT goldschmidtc randomrecursivetreesandthebolthausensznitmancoalescent AT martinj randomrecursivetreesandthebolthausensznitmancoalescent |