Attracting edge and strongly edge reinforced walks
The goal is to show that an edge-reinforced random, walk on a graph of bounded degree, with reinforcement weight function W taken from a general class of reciprocally summable reinforcement weight functions, traverses a random attracting edge at all large times. The statement of the main theorem is...
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| Format: | Journal article |
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2007
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| _version_ | 1797080876382683136 |
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| author | Limic, V Tarres, P |
| author_facet | Limic, V Tarres, P |
| author_sort | Limic, V |
| collection | OXFORD |
| description | The goal is to show that an edge-reinforced random, walk on a graph of bounded degree, with reinforcement weight function W taken from a general class of reciprocally summable reinforcement weight functions, traverses a random attracting edge at all large times. The statement of the main theorem is very close to settling a conjecture of Sellke [Technical Report 94-26 (1994) Purdue Univ.]. An important corollary of this main result says that if W is reciprocally summable and nondecreasing, the attracting edge exists on any graph of bounded degree, with probability 1. Another corollary is the main theorem of Limic [Ann. Probab. 31 (2003) 1615-1654], where the class of weights was restricted to reciprocally summable powers. The proof uses martingale and other techniques developed by the authors in separate studies of edge- and vertex-reinforced walks [Ann. Probab. 31 (2003) 1615-1654, Ann. Probab. 32 (2004) 2650-2701] and of nonconvergence properties of stochastic algorithms toward unstable equilibrium points of the associated deterministic dynamics [C. R. Acad. Sci. Sér. I Math. 330 (2000) 125-130]. © Institute of Mathematical Statistics, 2007. |
| first_indexed | 2024-03-07T01:06:31Z |
| format | Journal article |
| id | oxford-uuid:8b82583c-adb3-4060-9249-d407b66bcc55 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:06:31Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:8b82583c-adb3-4060-9249-d407b66bcc552022-03-26T22:38:39ZAttracting edge and strongly edge reinforced walksJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:8b82583c-adb3-4060-9249-d407b66bcc55Symplectic Elements at Oxford2007Limic, VTarres, PThe goal is to show that an edge-reinforced random, walk on a graph of bounded degree, with reinforcement weight function W taken from a general class of reciprocally summable reinforcement weight functions, traverses a random attracting edge at all large times. The statement of the main theorem is very close to settling a conjecture of Sellke [Technical Report 94-26 (1994) Purdue Univ.]. An important corollary of this main result says that if W is reciprocally summable and nondecreasing, the attracting edge exists on any graph of bounded degree, with probability 1. Another corollary is the main theorem of Limic [Ann. Probab. 31 (2003) 1615-1654], where the class of weights was restricted to reciprocally summable powers. The proof uses martingale and other techniques developed by the authors in separate studies of edge- and vertex-reinforced walks [Ann. Probab. 31 (2003) 1615-1654, Ann. Probab. 32 (2004) 2650-2701] and of nonconvergence properties of stochastic algorithms toward unstable equilibrium points of the associated deterministic dynamics [C. R. Acad. Sci. Sér. I Math. 330 (2000) 125-130]. © Institute of Mathematical Statistics, 2007. |
| spellingShingle | Limic, V Tarres, P Attracting edge and strongly edge reinforced walks |
| title | Attracting edge and strongly edge reinforced walks |
| title_full | Attracting edge and strongly edge reinforced walks |
| title_fullStr | Attracting edge and strongly edge reinforced walks |
| title_full_unstemmed | Attracting edge and strongly edge reinforced walks |
| title_short | Attracting edge and strongly edge reinforced walks |
| title_sort | attracting edge and strongly edge reinforced walks |
| work_keys_str_mv | AT limicv attractingedgeandstronglyedgereinforcedwalks AT tarresp attractingedgeandstronglyedgereinforcedwalks |