arXiv 1106.16632011-06-08math.COmath.PR

置换序列的置换正则性极限

Limits of permutation sequences through permutation regularity

Carlos Hoppen, Yoshiharu Kohayakawa, Carlos Gustavo Tamm de Araújo Moreira, Rudini Menezes Sampaio

摘要：

如果对于每个固定置换$τ$，序列元素中$τ$的出现密度收敛，则称置换序列$(σ_n)_{n \in \mathbb{N}}$收敛。我们证明了这样一个收敛序列有一个自然的极限对象，即勒贝格可测函数$Z:[0,1]^2 \to [0,1]$的附加属性，对于每一个固定的$x \in [0,1]$，限制$Z(x,\cdot)$是一个累积分布函数，对于每一个$y \in [0,1]$，限制$Z(\cdot,y)$满足“质量”条件。这个极限过程表现良好：极限对象类中的每个函数都是某个置换序列的极限，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。证明中的一个重要组成部分是一个新的随机排列模型，它概括了以前的模型，并且本身就很有趣。

英文摘要：

A permutation sequence $(σ_n)_{n \in \mathbb{N}}$ 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 $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative distribution function and, for every $y \in [0,1]$, the restriction $Z(\cdot,y)$ 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 important ingredient in the proofs is a new model of random permutations, which generalizes previous models and is interesting for its own sake.

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摘要：

英文摘要：

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