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 pi = pi(n), then the threshold is (Formula presented.). Our claim in one direction is in fact slightly stronger, namely, we show that if (Formula presented.) is sufficiently larger than (Formula presented.) 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 (Formula presented.) is sufficiently smaller than (Formula presented.), then asymptotically almost surely, the torus does not admit a topological embedding into Y.
Original language | English |
---|---|
Pages (from-to) | 664-675 |
Number of pages | 12 |
Journal | Random Structures and Algorithms |
Volume | 58 |
Issue number | 4 |
DOIs | |
State | Published - Jul 2021 |
Bibliographical note
Publisher Copyright:© 2021 Wiley Periodicals LLC.
Funding
information Leverhulme Trust
Funders | Funder number |
---|---|
Leverhulme Trust |
Keywords
- multiparameter threshold
- random simplicial complexes