Homological Connectivity Of Random 2-Complexes
Homological Connectivity Of Random 2-Complexes
复制标题
随机 2-复合体的同源连通性
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
R. Meshulam
中科院分区：
文献类型：
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作者：
N. Linial;R. Meshulam
设Δn−1表示（n−1）维单纯形。设Y为Δn−1的一个随机二维子复形，由Δn−1的完整一维骨架开始，然后以概率p独立地将每个2-单纯形相加得到。设$$
H_{1} {left( {Y;{Bbb F}_{2} }
ight)}
$$为Y的第一个模2系数同调群。对于任何趋于无穷的函数ω(n) $$
{mathop {lim }limits_{n o infty } }{kern 1pt} {kern 1pt} { ext{Prob}}{left[ {H_{1} {left( {Y;{Bbb F}_{2} }
ight)} = 0}
ight]} = left{ {egin{array}{*{20}c}
{{0p = frac{{2log n - omega {left( n
ight)}}}
{n}}} \
{{1p = frac{{2log n + omega {left( n
ight)}}}
{n}}} \
end{array} }
ight.
$$
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 $$
H_{1} {left( {Y;{Bbb F}_{2} }
ight)}
$$ denote the first homology group of Y with mod 2 coefficients. It is shown that for any function ω(n) that tends to infinity $$
{mathop {lim }limits_{n o infty } }{kern 1pt} {kern 1pt} { ext{Prob}}{left[ {H_{1} {left( {Y;{Bbb F}_{2} }
ight)} = 0}
ight]} = left{ {egin{array}{*{20}c}
{{0p = frac{{2log n - omega {left( n
ight)}}}
{n}}} \
{{1p = frac{{2log n + omega {left( n
ight)}}}
{n}}} \
end{array} }
ight.
$$