Loading…
Random walks avoiding their convex hull with a finite memory
Fix integers d≥2 and k≥d−1. Consider a random walk X0,X1,… in Rd in which, given X0,X1,…,Xn (n≥k), the next step Xn+1 is uniformly distributed on the unit ball centred at Xn, but conditioned that the line segment from Xn to Xn+1 intersects the convex hull of {0,Xn−k,…,Xn} only at Xn. For k=∞ this is...
Saved in:
| Published in: | Indagationes mathematicae 2020-01, Vol.31 (1), p.117-146 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| cited_by | cdi_FETCH-LOGICAL-c382t-fc93262b31dbfba224d78514dc58841b5fc1c627323f11204c06b5a43c114703 |
|---|---|
| cites | cdi_FETCH-LOGICAL-c382t-fc93262b31dbfba224d78514dc58841b5fc1c627323f11204c06b5a43c114703 |
| container_end_page | 146 |
| container_issue | 1 |
| container_start_page | 117 |
| container_title | Indagationes mathematicae |
| container_volume | 31 |
| creator | Comets, Francis Menshikov, Mikhail V. Wade, Andrew R. |
| description | Fix integers d≥2 and k≥d−1. Consider a random walk X0,X1,… in Rd in which, given X0,X1,…,Xn (n≥k), the next step Xn+1 is uniformly distributed on the unit ball centred at Xn, but conditioned that the line segment from Xn to Xn+1 intersects the convex hull of {0,Xn−k,…,Xn} only at Xn. For k=∞ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic, i.e., to have a limiting speed and a limiting direction. We establish ballisticity for the finite-k model, and comment on some open problems. In the case where d=2 and k=1, we obtain the limiting speed explicitly: it is 8∕(9π2). |
| doi_str_mv | 10.1016/j.indag.2019.11.002 |
| format | article |
| fullrecord | <record><control><sourceid>elsevier_hal_p</sourceid><recordid>TN_cdi_hal_primary_oai_HAL_hal_03179749v1</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0019357719300758</els_id><sourcerecordid>S0019357719300758</sourcerecordid><originalsourceid>FETCH-LOGICAL-c382t-fc93262b31dbfba224d78514dc58841b5fc1c627323f11204c06b5a43c114703</originalsourceid><addsrcrecordid>eNp9kE1LxDAQhoMouK7-Ai-5emjNJGnTgh6WxS9YEGTvIc3HNms_JKld99_bdcWjp2Fm3mdgHoSugaRAIL_dpr4zapNSAmUKkBJCT9AMCkGTHAg5RTMybRKWCXGOLmLcTq0gNJ-huzfVmb7FO9W8R6zG3hvfbfBQWx-w7rvRfuH6s2nwzg81Vtj5zg8Wt7btw_4SnTnVRHv1W-do_fiwXj4nq9enl-VilWhW0CFxumQ0pxUDU7lKUcqNKDLgRmdFwaHKnAadU8EocwCUcE3yKlOcaQAuCJujm-PZWjXyI_hWhb3slZfPi5U8zAgDUQpejjBl2TGrQx9jsO4PACIPruRW_riSB1cSQE6uJur-SNnpi9HbIKP2ttPW-GD1IE3v_-W_ATJacXA</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Random walks avoiding their convex hull with a finite memory</title><source>ScienceDirect Freedom Collection</source><creator>Comets, Francis ; Menshikov, Mikhail V. ; Wade, Andrew R.</creator><creatorcontrib>Comets, Francis ; Menshikov, Mikhail V. ; Wade, Andrew R.</creatorcontrib><description>Fix integers d≥2 and k≥d−1. Consider a random walk X0,X1,… in Rd in which, given X0,X1,…,Xn (n≥k), the next step Xn+1 is uniformly distributed on the unit ball centred at Xn, but conditioned that the line segment from Xn to Xn+1 intersects the convex hull of {0,Xn−k,…,Xn} only at Xn. For k=∞ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic, i.e., to have a limiting speed and a limiting direction. We establish ballisticity for the finite-k model, and comment on some open problems. In the case where d=2 and k=1, we obtain the limiting speed explicitly: it is 8∕(9π2).</description><identifier>ISSN: 0019-3577</identifier><identifier>EISSN: 1872-6100</identifier><identifier>DOI: 10.1016/j.indag.2019.11.002</identifier><language>eng</language><publisher>Elsevier B.V</publisher><subject>Ballistic ; Convex hull ; Mathematics ; Rancher ; Random walk ; Self-avoiding ; Speed</subject><ispartof>Indagationes mathematicae, 2020-01, Vol.31 (1), p.117-146</ispartof><rights>2019 Royal Dutch Mathematical Society (KWG)</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c382t-fc93262b31dbfba224d78514dc58841b5fc1c627323f11204c06b5a43c114703</citedby><cites>FETCH-LOGICAL-c382t-fc93262b31dbfba224d78514dc58841b5fc1c627323f11204c06b5a43c114703</cites><orcidid>0000-0002-3829-3406</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,314,780,784,885,27924,27925</link.rule.ids><backlink>$$Uhttps://hal.science/hal-03179749$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Comets, Francis</creatorcontrib><creatorcontrib>Menshikov, Mikhail V.</creatorcontrib><creatorcontrib>Wade, Andrew R.</creatorcontrib><title>Random walks avoiding their convex hull with a finite memory</title><title>Indagationes mathematicae</title><description>Fix integers d≥2 and k≥d−1. Consider a random walk X0,X1,… in Rd in which, given X0,X1,…,Xn (n≥k), the next step Xn+1 is uniformly distributed on the unit ball centred at Xn, but conditioned that the line segment from Xn to Xn+1 intersects the convex hull of {0,Xn−k,…,Xn} only at Xn. For k=∞ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic, i.e., to have a limiting speed and a limiting direction. We establish ballisticity for the finite-k model, and comment on some open problems. In the case where d=2 and k=1, we obtain the limiting speed explicitly: it is 8∕(9π2).</description><subject>Ballistic</subject><subject>Convex hull</subject><subject>Mathematics</subject><subject>Rancher</subject><subject>Random walk</subject><subject>Self-avoiding</subject><subject>Speed</subject><issn>0019-3577</issn><issn>1872-6100</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LxDAQhoMouK7-Ai-5emjNJGnTgh6WxS9YEGTvIc3HNms_JKld99_bdcWjp2Fm3mdgHoSugaRAIL_dpr4zapNSAmUKkBJCT9AMCkGTHAg5RTMybRKWCXGOLmLcTq0gNJ-huzfVmb7FO9W8R6zG3hvfbfBQWx-w7rvRfuH6s2nwzg81Vtj5zg8Wt7btw_4SnTnVRHv1W-do_fiwXj4nq9enl-VilWhW0CFxumQ0pxUDU7lKUcqNKDLgRmdFwaHKnAadU8EocwCUcE3yKlOcaQAuCJujm-PZWjXyI_hWhb3slZfPi5U8zAgDUQpejjBl2TGrQx9jsO4PACIPruRW_riSB1cSQE6uJur-SNnpi9HbIKP2ttPW-GD1IE3v_-W_ATJacXA</recordid><startdate>202001</startdate><enddate>202001</enddate><creator>Comets, Francis</creator><creator>Menshikov, Mikhail V.</creator><creator>Wade, Andrew R.</creator><general>Elsevier B.V</general><general>Elsevier</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><orcidid>https://orcid.org/0000-0002-3829-3406</orcidid></search><sort><creationdate>202001</creationdate><title>Random walks avoiding their convex hull with a finite memory</title><author>Comets, Francis ; Menshikov, Mikhail V. ; Wade, Andrew R.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c382t-fc93262b31dbfba224d78514dc58841b5fc1c627323f11204c06b5a43c114703</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Ballistic</topic><topic>Convex hull</topic><topic>Mathematics</topic><topic>Rancher</topic><topic>Random walk</topic><topic>Self-avoiding</topic><topic>Speed</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Comets, Francis</creatorcontrib><creatorcontrib>Menshikov, Mikhail V.</creatorcontrib><creatorcontrib>Wade, Andrew R.</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Indagationes mathematicae</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Comets, Francis</au><au>Menshikov, Mikhail V.</au><au>Wade, Andrew R.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Random walks avoiding their convex hull with a finite memory</atitle><jtitle>Indagationes mathematicae</jtitle><date>2020-01</date><risdate>2020</risdate><volume>31</volume><issue>1</issue><spage>117</spage><epage>146</epage><pages>117-146</pages><issn>0019-3577</issn><eissn>1872-6100</eissn><abstract>Fix integers d≥2 and k≥d−1. Consider a random walk X0,X1,… in Rd in which, given X0,X1,…,Xn (n≥k), the next step Xn+1 is uniformly distributed on the unit ball centred at Xn, but conditioned that the line segment from Xn to Xn+1 intersects the convex hull of {0,Xn−k,…,Xn} only at Xn. For k=∞ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic, i.e., to have a limiting speed and a limiting direction. We establish ballisticity for the finite-k model, and comment on some open problems. In the case where d=2 and k=1, we obtain the limiting speed explicitly: it is 8∕(9π2).</abstract><pub>Elsevier B.V</pub><doi>10.1016/j.indag.2019.11.002</doi><tpages>30</tpages><orcidid>https://orcid.org/0000-0002-3829-3406</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0019-3577 |
| ispartof | Indagationes mathematicae, 2020-01, Vol.31 (1), p.117-146 |
| issn | 0019-3577 1872-6100 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_03179749v1 |
| source | ScienceDirect Freedom Collection |
| subjects | Ballistic Convex hull Mathematics Rancher Random walk Self-avoiding Speed |
| title | Random walks avoiding their convex hull with a finite memory |
| url | http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-05T06%3A56%3A52IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-elsevier_hal_p&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Random%20walks%20avoiding%20their%20convex%20hull%20with%20a%20finite%20memory&rft.jtitle=Indagationes%20mathematicae&rft.au=Comets,%20Francis&rft.date=2020-01&rft.volume=31&rft.issue=1&rft.spage=117&rft.epage=146&rft.pages=117-146&rft.issn=0019-3577&rft.eissn=1872-6100&rft_id=info:doi/10.1016/j.indag.2019.11.002&rft_dat=%3Celsevier_hal_p%3ES0019357719300758%3C/elsevier_hal_p%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c382t-fc93262b31dbfba224d78514dc58841b5fc1c627323f11204c06b5a43c114703%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_id=info:pmid/&rfr_iscdi=true |