Let be the set of all mappings . The corresponding graph of is a union of disjoint connected unicyclic components. We assume that each is chosen uniformly at random (i.e., with probability ). The cycle of contained within its largest component is called the one. For any , let denote the length of this cycle. In this paper, we establish the convergence in distribution of and find the limits of its expectation and variance as . For large enough, we also show that nearly 55% of all cyclic vertices of a random mapping lie in its deepest cycle and that a vertex from the longest cycle of does not belong to its largest component with approximate probability 0.075.

中文翻译：

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关于随机映射的最深循环

让 是所有映射的集合。的相应图是不相交连接的单环分量的并集。我们假设每个都是随机均匀选择的（即概率为 ）。其最大组成部分所包含的循环称为一循环。对于任意 ，让 表示该循环的长度。在本文中，我们建立了分布的收敛性，并找到其期望和方差的极限为 。对于足够大的情况，我们还表明，随机映射的所有循环顶点中近 55% 位于其最深循环中，并且最长循环中的顶点不属于其最大分量，概率约为 0.075。