Topological embeddings into random 2‐complexes. Issue 4 (6th January 2021) - Farber, Michael; Nowik, Tahl
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 .