2,190 research outputs found
The threshold for integer homology in random d-complexes
Let Y ~ Y_d(n,p) denote the Bernoulli random d-dimensional simplicial
complex. We answer a question of Linial and Meshulam from 2003, showing that
the threshold for vanishing of homology H_{d-1}(Y; Z) is less than 80d log n /
n. This bound is tight, up to a constant factor.Comment: 12 pages, updated to include an additional torsion group boun
Operators on random hypergraphs and random simplicial complexes
Random hypergraphs and random simplicial complexes have potential
applications in computer science and engineering. Various models of random
hypergraphs and random simplicial complexes on n-points have been studied. Let
L be a simplicial complex. In this paper, we study random sub-hypergraphs and
random sub-complexes of L. By considering the minimal complex that a
sub-hypergraph can be embedded in and the maximal complex that can be embedded
in a sub-hypergraph, we define some operators on the space of probability
functions on sub-hypergraphs of L. We study the compositions of these operators
as well as their actions on the space of probability functions. As applications
in computer science, we give algorithms generating large sparse random
hypergraphs and large sparse random simplicial complexes.Comment: 22 page
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