Topological embeddings into random 2-complexes

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 p0p13p22=1n. Our claim in one direction is in fact slightly stronger, namely, we show that if p0p13p22 is sufficiently larger than 1n 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 p0p13p22 is sufficiently smaller than 1n, then asymptotically almost surely, the torus does not admit a topological embedding into Y.