Note: The neighborhood complex of a random graph
Journal of Combinatorial Theory Series A
Minors in random and expanding hypergraphs
Proceedings of the twenty-seventh annual symposium on Computational geometry
On multiplicative λ-approximations and some geometric applications
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On laplacians of random complexes
Proceedings of the twenty-eighth annual symposium on Computational geometry
Topics of Stochastic Algebraic Topology
Electronic Notes in Theoretical Computer Science (ENTCS)
High dimensional expanders and property testing
Proceedings of the 5th conference on Innovations in theoretical computer science
Quantum Information & Computation
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Let Δn−1 denote the (n − 1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of Δn−1 obtained by starting with the full 1-dimensional skeleton of Δn−1 and then adding each 2−simplex independently with probability p. Let $$H_{1} {\left( {Y;{\Bbb F}_{2} } \right)}$$ denote the first homology group of Y with mod 2 coefficients. It is shown that for any function ω(n) that tends to infinity $${\mathop {\lim }\limits_{n \to \infty } }{\kern 1pt} {\kern 1pt} {\text{Prob}}{\left[ {H_{1} {\left( {Y;{\Bbb F}_{2} } \right)} = 0} \right]} = \left\{ {\begin{array}{*{20}c}{{0p = \frac{{2\log n - \omega {\left( n \right)}}}{n}}} \\ {{1p = \frac{{2\log n + \omega {\left( n \right)}}}{n}}} \\ \end{array} } \right.$$