论文标题
摩洛槌随机排列中的循环
Cycles in Mallows random permutations
论文作者
论文摘要
我们研究$ 1，\ dots，n $随机绘制的周期计数。在此分布下，选择每个排列$π\ in s_n $中的概率与$ q^{\ text {inv}（π）} $成正比，其中$ q> 0 $是一个参数，$ \ text {inv}（π）$表示$π$的反转数。对于$ \ ell $固定，我们研究向量$（c_1（π_n），\ dots，c_ \ ell（π_n））$其中$ c_i（π）$表示$π$中的长度$ i $ in $ i $ in $π$和$π_n$的循环的数量。 在这里我们表明，如果$ 0 <q <1 $是固定的，并且$ n \ to \ infty $，则有正常数$ m_i $，因此每个$ c_i（π_n）$的每个$（1+o（1）\ cdot m_i \ cdot m_i \ cdot n $，并且循环计数的量很大程度上可以重新分配到凝分级别的分配。我们的结果还表明，当$ q> 1 $时，均匀循环的行为与奇数周期之间存在明显的区别。均匀的循环计数仍然具有线性均值，并且正确重新缩放后倾向于多元高斯分布。另一方面，对于奇数周期的计算，限制行为取决于$ q> 1 $的奇偶校验。 Both $(C_1(Π_{2n}),C_3(Π_{2n}),\dots)$ and $(C_1(Π_{2n+1}),C_3(Π_{2n+1}),\dots)$ have discrete limiting distributions -- they do not need to be renormalized -- but the two limiting distributions are distinct for all $q>1$.我们用Gnedin和Olshanski对木棍模型的Bi-Infinite扩展来描述这些限制分布。 我们还研究了这些限制分布，并研究了高斯极限定律所涉及的常数的行为。例如，我们表明，作为$ q \ downarrow 1 $，预期的1循环数量趋向于$ 1/2 $ - 奇怪的是，这与对应于$ q = 1 $的值不同。此外，我们在限制概率措施中表现出有趣的“振荡”行为，$ q>> 1 $和$ n $奇数与$ n $偶数。
We study cycle counts in permutations of $1,\dots,n$ drawn at random according to the Mallows distribution. Under this distribution, each permutation $π\in S_n$ is selected with probability proportional to $q^{\text{inv}(π)}$, where $q>0$ is a parameter and $\text{inv}(π)$ denotes the number of inversions of $π$. For $\ell$ fixed, we study the vector $(C_1(Π_n),\dots,C_\ell(Π_n))$ where $C_i(π)$ denotes the number of cycles of length $i$ in $π$ and $Π_n$ is sampled according to the Mallows distribution. Here we show that if $0<q<1$ is fixed and $n\to\infty$ then there are positive constants $m_i$ such that each $C_i(Π_n)$ has mean $(1+o(1)) \cdot m_i\cdot n$ and the vector of cycle counts can be suitably rescaled to tend to a joint Gaussian distribution. Our results also show that when $q>1$ there is striking difference between the behaviour of the even and the odd cycles. The even cycle counts still have linear means, and when properly rescaled tend to a multivariate Gaussian distribution. For the odd cycle counts on the other hand, the limiting behaviour depends on the parity of $n$ when $q>1$. Both $(C_1(Π_{2n}),C_3(Π_{2n}),\dots)$ and $(C_1(Π_{2n+1}),C_3(Π_{2n+1}),\dots)$ have discrete limiting distributions -- they do not need to be renormalized -- but the two limiting distributions are distinct for all $q>1$. We describe these limiting distributions in terms of Gnedin and Olshanski's bi-infinite extension of the Mallows model. We also investigate these limiting distributions, and study the behaviour of the constants involved in the Gaussian limit laws. We for example show that as $q\downarrow 1$ the expected number of 1-cycles tends to $1/2$ -- which, curiously, differs from the value corresponding to $q=1$. In addition we exhibit an interesting "oscillating" behaviour in the limiting probability measures for $q>1$ and $n$ odd versus $n$ even.
