{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T13:48:13Z","timestamp":1740491293625,"version":"3.38.0"},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2014,9]]},"abstract":"<jats:p>Suppose that red and blue points occur in<jats:bold>R<\/jats:bold><jats:sup><jats:italic>d<\/jats:italic><\/jats:sup>according to two simple point processes with finite intensities \u03bb<jats:sub><jats:italic>R<\/jats:italic><\/jats:sub>and \u03bb<jats:sub><jats:italic>B<\/jats:italic><\/jats:sub>, respectively. Furthermore, let \u03bd and \u03bc be two probability distributions on the strictly positive integers with means \u03bd\u0305 and \u03bc\u0305, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law \u03bd (\u03bc). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law \u03bd or \u03bc depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if \u03bb<jats:sub><jats:italic>R<\/jats:italic><\/jats:sub>\u03bd\u0305 = \u03bb<jats:sub><jats:italic>B<\/jats:italic><\/jats:sub>\u03bc\u0305, including the case when \u03bd\u0305 = \u03bc\u0305 = \u221e so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on \u03bd and \u03bc for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and H\u00e4ggstr\u00f6m.<\/jats:p>","DOI":"10.1017\/s0021900200011669","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T10:50:00Z","timestamp":1459248600000},"page":"769-779","source":"Crossref","is-referenced-by-count":0,"title":["Invariant Bipartite Random Graphs on Rd"],"prefix":"10.1017","volume":"51","author":[{"given":"Fabio","family":"Lopes","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,2,19]]},"reference":[{"volume-title":"Random Graph Dynamics","year":"2007","key":"S0021900200011669_ref8"},{"key":"S0021900200011669_ref10","doi-asserted-by":"publisher","DOI":"10.2307\/2312726"},{"key":"S0021900200011669_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s11512-010-0139-8"},{"key":"S0021900200011669_ref5","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1151337072"},{"key":"S0021900200011669_ref17","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1177004978"},{"key":"S0021900200011669_ref4","first-page":"583","volume":"18","year":"2012","journal-title":"Markov Process. 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