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Topology"],"published-print":{"date-parts":[[2025,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We consider face and cycle percolation as models for continuum percolation based on random simplicial complexes in Euclidean space. More precisely, we consider the Vietoris\u2013Rips with connection radius 1, with respect to Euclidean norm, on a stationary Poisson process in <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathbb {R}}^D$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>D<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> of intensity <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\lambda $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bb<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. Face percolation is defined through infinite sequences of <jats:italic>d<\/jats:italic>-simplices sharing a <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$(d-1)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-dimensional face. In contrast, cycle percolation demands the existence of infinite <jats:italic>d<\/jats:italic>-cycles, thereby generalizing the lattice notion of plaquette percolation. We discuss the sharp phase transition for face percolation and derive comparison results between the critical intensities for face and cycle percolation. Finally, we consider an alternate version of simplex percolation, by declaring simplices to be neighbors whenever they are sufficiently close to each other, and prove a strict inequality involving the critical intensity of this alternate version and that of face percolation.<\/jats:p>","DOI":"10.1007\/s41468-025-00202-2","type":"journal-article","created":{"date-parts":[[2025,2,4]],"date-time":"2025-02-04T21:11:14Z","timestamp":1738703474000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Face and cycle percolation"],"prefix":"10.1007","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4136-3740","authenticated-orcid":false,"given":"Christian","family":"Hirsch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Valesin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,2,4]]},"reference":[{"key":"202_CR1","unstructured":"Balister, P., Bollob\u00e1s, B., Riordan, O.: Essential enhancements revisited (2014). arXiv preprint arXiv:1402.0834"},{"key":"202_CR2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/EJP.v18-2468","volume":"18","author":"B B\u0142aszczyszyn","year":"2013","unstructured":"B\u0142aszczyszyn, B., Yogeshwaran, D.: Clustering and percolation of point processes. 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