Limits of permutation sequences
Limits of permutation sequences
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排列序列的极限
DOI：
10.1016/j.jctb.2012.09.003
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
R. Sampaio
中科院分区：
文献类型：
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作者：
C. Hoppen;Y. Kohayakawa;Carlos Gustavo T. de A. Moreira;Balázs Ráth;R. Sampaio
如果对于每个固定排列 τ，序列元素中 τ 出现的密度收敛，则称排列序列 [公式：参见文本] 是收敛的。我们证明这样的收敛序列有一个自然极限对象，即勒贝格可测函数 Z:[0,1]2→[0,1]，其附加属性是，对于每个固定的 x∈[0,1]，限制 Z(x,⋅) 是累积分布函数，并且对于每个 y∈[0,1]，限制 Z(⋅,y) 满足“质量”条件。这个极限过程是良好的：极限对象类中的每个函数都是某个排列序列的极限，并且其中两个函数是同一序列的极限当且仅当它们几乎处处相等。证明中的一个要素是随机排列的新模型，它概括了以前的模型，并且本身可能很有趣。
A permutation sequence [Formula: see text] is said to be convergent if, for every fixed permutation τ, the density of occurrences of τ in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function Z:[0,1]2→[0,1] with the additional properties that, for every fixed x∈[0,1], the restriction Z(x,⋅) is a cumulative distribution function and, for every y∈[0,1], the restriction Z(⋅,y) satisfies a “mass” condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An ingredient in the proofs is a new model of random permutations, which generalizes previous models and might be interesting for its own sake.