Homological Connectivity Of Random 2-Complexes

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

$$ {\mathop {\lim }\limits_{n \to \infty } }{\kern 1pt} {\kern 1pt} {\text{Prob}}{\left[ {H_{1} {\left( {Y;{\Bbb F}_{2} } \right)} = 0} \right]} = \left\{ {\begin{array}{*{20}c} {{0p = \frac{{2\log n - \omega {\left( n \right)}}} {n}}} \\ {{1p = \frac{{2\log n + \omega {\left( n \right)}}} {n}}} \\ \end{array} } \right. $$

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Correspondence to Nathan Linial*.

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* Supported by an Israel Science Foundation grant.

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Linial*, N., Meshulam*, R. Homological Connectivity Of Random 2-Complexes. Combinatorica 26, 475–487 (2006). https://doi.org/10.1007/s00493-006-0027-9

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Mathematics Subject Classification (2000):

  • 55U10
  • 05C80