Topological embeddings into random 2‐complexes
Michael Färber, Tahl Nowik · Random Structures and Algorithms · 2021
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 . Our claim in one direction is in fact slightly stronger, namely, we show that if is sufficiently larger than 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 is sufficiently smaller than , then asymptotically almost surely, the torus does not admit a topological embedding into Y.