Random Čech complexes on manifolds with boundary
Let M be a compact, unit volume, Riemannian manifold with boundary. We study the homology of a random Čech‐complex generated by a homogeneous Poisson process in M. Our main results are two asymptotic threshold formulas, an upper threshold above which the Čech complex recovers the kth homology of M w...
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| Veröffentlicht in: | Random structures & algorithms 2022-09, Vol.61 (2), p.309-352 |
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| creator | Kergorlay, Henry‐Louis Tillmann, Ulrike Vipond, Oliver |
| description | Let M be a compact, unit volume, Riemannian manifold with boundary. We study the homology of a random Čech‐complex generated by a homogeneous Poisson process in M. Our main results are two asymptotic threshold formulas, an upper threshold above which the Čech complex recovers the kth homology of M
with high probability, and a lower threshold below which it almost certainly does not. These thresholds share the same leading term. This extends work of Bobrowski–Weinberger and Bobrowski–Oliveira who establish similar formulas when M has no boundary. The cases with and without boundary differ: the corresponding common leading terms for the upper and lower thresholds differ being log(n) when M is closed and (2−2/d)log(n) when M has boundary; here n is the expected number of sample points. Our analysis identifies a special type of homological cycle occurring close to the boundary. |
| doi_str_mv | 10.1002/rsa.21062 |
| format | Article |
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with high probability, and a lower threshold below which it almost certainly does not. These thresholds share the same leading term. This extends work of Bobrowski–Weinberger and Bobrowski–Oliveira who establish similar formulas when M has no boundary. The cases with and without boundary differ: the corresponding common leading terms for the upper and lower thresholds differ being log(n) when M is closed and (2−2/d)log(n) when M has boundary; here n is the expected number of sample points. Our analysis identifies a special type of homological cycle occurring close to the boundary.</description><identifier>ISSN: 1042-9832</identifier><identifier>EISSN: 1098-2418</identifier><identifier>DOI: 10.1002/rsa.21062</identifier><language>eng</language><publisher>New York: John Wiley & Sons, Inc</publisher><subject>Homology ; random geometric complexes ; Riemann manifold ; Statistical analysis ; stochastic topology ; Thresholds</subject><ispartof>Random structures & algorithms, 2022-09, Vol.61 (2), p.309-352</ispartof><rights>2022 The Authors. Random Structures & Algorithms published by Wiley Periodicals LLC.</rights><rights>2022. This article is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c3322-db2dc8d93e3a5ff0dbc698ba67df55cbbde4240e108de796d9e3df2d568e2d8a3</citedby><cites>FETCH-LOGICAL-c3322-db2dc8d93e3a5ff0dbc698ba67df55cbbde4240e108de796d9e3df2d568e2d8a3</cites><orcidid>0000-0003-0520-3803 ; 0000-0003-2355-5570 ; 0000-0002-8076-7660</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://onlinelibrary.wiley.com/doi/pdf/10.1002%2Frsa.21062$$EPDF$$P50$$Gwiley$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://onlinelibrary.wiley.com/doi/full/10.1002%2Frsa.21062$$EHTML$$P50$$Gwiley$$Hfree_for_read</linktohtml><link.rule.ids>314,777,781,1412,27905,27906,45555,45556</link.rule.ids></links><search><creatorcontrib>Kergorlay, Henry‐Louis</creatorcontrib><creatorcontrib>Tillmann, Ulrike</creatorcontrib><creatorcontrib>Vipond, Oliver</creatorcontrib><title>Random Čech complexes on manifolds with boundary</title><title>Random structures & algorithms</title><description>Let M be a compact, unit volume, Riemannian manifold with boundary. We study the homology of a random Čech‐complex generated by a homogeneous Poisson process in M. Our main results are two asymptotic threshold formulas, an upper threshold above which the Čech complex recovers the kth homology of M
with high probability, and a lower threshold below which it almost certainly does not. These thresholds share the same leading term. This extends work of Bobrowski–Weinberger and Bobrowski–Oliveira who establish similar formulas when M has no boundary. The cases with and without boundary differ: the corresponding common leading terms for the upper and lower thresholds differ being log(n) when M is closed and (2−2/d)log(n) when M has boundary; here n is the expected number of sample points. Our analysis identifies a special type of homological cycle occurring close to the boundary.</description><subject>Homology</subject><subject>random geometric complexes</subject><subject>Riemann manifold</subject><subject>Statistical analysis</subject><subject>stochastic topology</subject><subject>Thresholds</subject><issn>1042-9832</issn><issn>1098-2418</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><sourceid>24P</sourceid><sourceid>WIN</sourceid><recordid>eNp1kMtOwzAQRS0EEuWx4A8isWKR1h4nrr2sKl5SJaQCa8vx2GqqJC52q9J_4LP4MFLCltXcxZm5o0PIDaNjRilMYjJjYFTACRkxqmQOBZOnx1xAriSHc3KR0ppSOuXAR4QtTYehzb6_nF1lNrSbxn26lIUua01X-9Bgyvb1dpVVYdehiYcrcuZNk9z137wk7w_3b_OnfPHy-DyfLXLLOUCOFaCVqLjjpvSeYmWFkpURU_RlaasKXQEFdYxKdFMlUDmOHrAU0gFKwy_J7XB3E8PHzqWtXodd7PpKDUJRBZIJ2VN3A2VjSCk6rzexbvs3NaP6aET3RvSvkZ6dDOy-btzhf1AvX2fDxg8BHGLq</recordid><startdate>202209</startdate><enddate>202209</enddate><creator>Kergorlay, Henry‐Louis</creator><creator>Tillmann, Ulrike</creator><creator>Vipond, Oliver</creator><general>John Wiley & Sons, Inc</general><general>Wiley Subscription Services, Inc</general><scope>24P</scope><scope>WIN</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0000-0003-0520-3803</orcidid><orcidid>https://orcid.org/0000-0003-2355-5570</orcidid><orcidid>https://orcid.org/0000-0002-8076-7660</orcidid></search><sort><creationdate>202209</creationdate><title>Random Čech complexes on manifolds with boundary</title><author>Kergorlay, Henry‐Louis ; Tillmann, Ulrike ; Vipond, Oliver</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c3322-db2dc8d93e3a5ff0dbc698ba67df55cbbde4240e108de796d9e3df2d568e2d8a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Homology</topic><topic>random geometric complexes</topic><topic>Riemann manifold</topic><topic>Statistical analysis</topic><topic>stochastic topology</topic><topic>Thresholds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kergorlay, Henry‐Louis</creatorcontrib><creatorcontrib>Tillmann, Ulrike</creatorcontrib><creatorcontrib>Vipond, Oliver</creatorcontrib><collection>Wiley-Blackwell Open Access Titles</collection><collection>Wiley Free Content</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Random structures & algorithms</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kergorlay, Henry‐Louis</au><au>Tillmann, Ulrike</au><au>Vipond, Oliver</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Random Čech complexes on manifolds with boundary</atitle><jtitle>Random structures & algorithms</jtitle><date>2022-09</date><risdate>2022</risdate><volume>61</volume><issue>2</issue><spage>309</spage><epage>352</epage><pages>309-352</pages><issn>1042-9832</issn><eissn>1098-2418</eissn><abstract>Let M be a compact, unit volume, Riemannian manifold with boundary. We study the homology of a random Čech‐complex generated by a homogeneous Poisson process in M. Our main results are two asymptotic threshold formulas, an upper threshold above which the Čech complex recovers the kth homology of M
with high probability, and a lower threshold below which it almost certainly does not. These thresholds share the same leading term. This extends work of Bobrowski–Weinberger and Bobrowski–Oliveira who establish similar formulas when M has no boundary. The cases with and without boundary differ: the corresponding common leading terms for the upper and lower thresholds differ being log(n) when M is closed and (2−2/d)log(n) when M has boundary; here n is the expected number of sample points. Our analysis identifies a special type of homological cycle occurring close to the boundary.</abstract><cop>New York</cop><pub>John Wiley & Sons, Inc</pub><doi>10.1002/rsa.21062</doi><tpages>87</tpages><orcidid>https://orcid.org/0000-0003-0520-3803</orcidid><orcidid>https://orcid.org/0000-0003-2355-5570</orcidid><orcidid>https://orcid.org/0000-0002-8076-7660</orcidid><oa>free_for_read</oa></addata></record> |
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| subjects | Homology random geometric complexes Riemann manifold Statistical analysis stochastic topology Thresholds |
| title | Random Čech complexes on manifolds with boundary |
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