Linear embeddings of random complexes

Abstract

For $X\sim X(n;1,n^{-{\alpha }_1},n^{-{\alpha }_2},...)$ in the multiparameter random simplicial complex model we establish necessary and sufficient strict inequalities on the ${\alpha }_i$'s to linearly embed the complex into ${\mathbb{R}}^{2d}$.

15 pages. There was an error for the odd-dimensional case in the previous version. This version proves the threshold for embeddability into even dimensions only. Section 2 has also been expanded to include more detail