Bindweeds or random walks in random environments on multiplexed trees and their asympotics
We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree.The term multiplexed means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal deg...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2003-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/3324/pdf |
| _version_ | 1797270638986002432 |
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| author | Mikhail Menshikov Dimitri Petritis Serguei Popov |
| author_facet | Mikhail Menshikov Dimitri Petritis Serguei Popov |
| author_sort | Mikhail Menshikov |
| collection | DOAJ |
| description | We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree.The term multiplexed means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\} × \{1,...,d\}.$ This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term random environment means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates.This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (i.e.the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere. |
| first_indexed | 2024-04-25T02:07:28Z |
| format | Article |
| id | doaj.art-93b48e1312f243769e279695e4765e48 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:07:28Z |
| publishDate | 2003-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-93b48e1312f243769e279695e4765e482024-03-07T14:29:56ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502003-01-01DMTCS Proceedings vol. AC,...Proceedings10.46298/dmtcs.33243324Bindweeds or random walks in random environments on multiplexed trees and their asympoticsMikhail Menshikov0Dimitri Petritis1https://orcid.org/0000-0003-0082-9049Serguei Popov2Department of Mathematical SciencesInstitut de Recherche Mathématique de RennesInstituto de MatematicaWe report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree.The term multiplexed means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\} × \{1,...,d\}.$ This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term random environment means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates.This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (i.e.the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere.https://dmtcs.episciences.org/3324/pdfmatrix multiplicative cascadesmarkov chaintreesrandom environmentrecurrence criteria[info.info-ds] computer science [cs]/data structures and algorithms [cs.ds][info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co][info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| spellingShingle | Mikhail Menshikov Dimitri Petritis Serguei Popov Bindweeds or random walks in random environments on multiplexed trees and their asympotics Discrete Mathematics & Theoretical Computer Science matrix multiplicative cascades markov chain trees random environment recurrence criteria [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| title | Bindweeds or random walks in random environments on multiplexed trees and their asympotics |
| title_full | Bindweeds or random walks in random environments on multiplexed trees and their asympotics |
| title_fullStr | Bindweeds or random walks in random environments on multiplexed trees and their asympotics |
| title_full_unstemmed | Bindweeds or random walks in random environments on multiplexed trees and their asympotics |
| title_short | Bindweeds or random walks in random environments on multiplexed trees and their asympotics |
| title_sort | bindweeds or random walks in random environments on multiplexed trees and their asympotics |
| topic | matrix multiplicative cascades markov chain trees random environment recurrence criteria [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| url | https://dmtcs.episciences.org/3324/pdf |
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