Topological embeddings into random 2‐complexes. Issue 4 (6th January 2021)

• Record Type: Journal Article
• Authors: Farber, Michael; Nowik, Tahl
• Is Part Of: Random structures & algorithms. Volume 58:Issue 4(2021)
• Page Start: 664
• Page End: 675
• Publication Date: 2021-01-06
• Subjects: multiparameter threshold -- random simplicial complexes; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20987
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: We consider 2‐dimensional random simplicial complexes Y in the multiparameter model. We establish the multiparameter threshold for the property that every 2‐dimensional simplicial complex S admits a topological embedding into Y asymptotically almost surely. Namely, if in the procedure of the multiparameter model on n vertices, each i ‐dimensional simplex is taken with probability p i = p i ( n ), then the threshold is p 0 p 1 3 p 2 2 = 1 n . Our claim in one direction is in fact slightly stronger, namely, we show that if p 0 p 1 3 p 2 2 is sufficiently larger than 1 n then every S has a fixed subdivision S′ which admits a simplicial embedding into Y asymptotically almost surely. In the other direction we show that if p 0 p 1 3 p 2 2 is sufficiently smaller than 1 n, then asymptotically almost surely, the torus does not admit a topological embedding into Y .