{"id":340136,"date":"2016-12-21T14:31:06","date_gmt":"2016-12-21T22:31:06","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&#038;p=340136"},"modified":"2018-10-16T21:20:33","modified_gmt":"2018-10-17T04:20:33","slug":"percolation-invariant-poisson-graphs-d-degrees","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/percolation-invariant-poisson-graphs-d-degrees\/","title":{"rendered":"Percolation in invariant Poisson graphs with i.i.d. degrees"},"content":{"rendered":"<p>Let each point of a homogeneous Poisson process in R<em><sup>d<\/sup><\/em> independently be equipped with a random number of stubs (half-edges) according to a given probability distribution <em>\u00b5<\/em> on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution <em>\u00b5<\/em>. Leaving aside degenerate cases, we prove that for any <em>\u00b5<\/em> there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on <em>\u00b5<\/em> for the absence and presence of infinite components.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let each point of a homogeneous Poisson process in Rd independently be equipped with a random number of stubs (half-edges) according to a given probability distribution \u00b5 on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution \u00b5. Leaving aside degenerate cases, we prove [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Springer 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