{"title":"反身巴拿赫空间中的随机几何图形","authors":"József Balogh, Mark Walters, András Zsák","doi":"arxiv-2409.04237","DOIUrl":null,"url":null,"abstract":"We investigate a random geometric graph model introduced by Bonato and\nJanssen. The vertices are the points of a countable dense set $S$ in a\n(necessarily separable) normed vector space $X$, and each pair of points are\njoined independently with some fixed probability $p$ (with $0<p<1$) if they are\nless than distance $1$ apart. A countable dense set $S$ in a normed space is\nRado, if the resulting graph is almost surely unique up to isomorphism: that is\nany two such graphs are, almost surely, isomorphic. Not surprisingly, understanding which sets are Rado is closely related to the\ngeometry of the underlying normed space. It turns out that a key question is in\nwhich spaces must step-isometries (maps that preserve the integer parts of\ndistances) on dense subsets necessarily be isometries. We answer this question\nfor a large class of Banach spaces including all strictly convex reflexive\nspaces. In the process we prove results on the interplay between the norm\ntopology and weak topology that may be of independent interest. As a consequence of these Banach space results we show that almost all\ncountable dense sets in strictly convex reflexive spaces are strongly non-Rado\n(that is, any two graphs are almost surely non-isomorphic). However, we show\nthat there do exist Rado sets even in $\\ell_2$. Finally we construct a Banach\nspaces in which all countable dense set are strongly non-Rado.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Random Geometric Graphs in Reflexive Banach Spaces\",\"authors\":\"József Balogh, Mark Walters, András Zsák\",\"doi\":\"arxiv-2409.04237\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We investigate a random geometric graph model introduced by Bonato and\\nJanssen. The vertices are the points of a countable dense set $S$ in a\\n(necessarily separable) normed vector space $X$, and each pair of points are\\njoined independently with some fixed probability $p$ (with $0<p<1$) if they are\\nless than distance $1$ apart. A countable dense set $S$ in a normed space is\\nRado, if the resulting graph is almost surely unique up to isomorphism: that is\\nany two such graphs are, almost surely, isomorphic. Not surprisingly, understanding which sets are Rado is closely related to the\\ngeometry of the underlying normed space. It turns out that a key question is in\\nwhich spaces must step-isometries (maps that preserve the integer parts of\\ndistances) on dense subsets necessarily be isometries. We answer this question\\nfor a large class of Banach spaces including all strictly convex reflexive\\nspaces. In the process we prove results on the interplay between the norm\\ntopology and weak topology that may be of independent interest. As a consequence of these Banach space results we show that almost all\\ncountable dense sets in strictly convex reflexive spaces are strongly non-Rado\\n(that is, any two graphs are almost surely non-isomorphic). However, we show\\nthat there do exist Rado sets even in $\\\\ell_2$. Finally we construct a Banach\\nspaces in which all countable dense set are strongly non-Rado.\",\"PeriodicalId\":501036,\"journal\":{\"name\":\"arXiv - MATH - Functional Analysis\",\"volume\":\"10 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Functional Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.04237\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Functional Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.04237","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Random Geometric Graphs in Reflexive Banach Spaces
We investigate a random geometric graph model introduced by Bonato and
Janssen. The vertices are the points of a countable dense set $S$ in a
(necessarily separable) normed vector space $X$, and each pair of points are
joined independently with some fixed probability $p$ (with $0
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