Cycle structure of random permutations with cycle weights

On the Cycle Structure of Mallows Permutations

We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $π\in \mathbb{S}_n$ is proportional to $q^{\textrm{inv}(π)}$ where $0<q\le 1$ and $\textrm{inv}(π)$ is the number of inversions in $π$. We show that the expected length of the cycle containing a given point is of order $\min\{(1-q)^{-2}, n\}$. This marks the existence of two asymptotic regimes: with high probability, when $n$ tends to infinity with $(1-q)^{-2} \ll n$ then all cycles have size $o(n)$ whereas when $n$ tends to infinity with $(1-q)^{-…

Profiles of permutations

This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $σ$ and, on the other hand, permutations selected according to the Ewens distribution with parameter $σ$. In particular we show that the asymptotic expected number of cycles of random permutations of $[n]$ with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter 1/2 are all ${1 \over 2} \log n + O(1)$, and the variance is of the same order. Furthermore, we show that in perm…

随机标准化排列的循环结构

Cycle structure of random standardized permutations

本文研究随机标准化排列模型，分析不同长度循环数的联合分布，揭示小循环和大循环的极限分布，证明总循环数的中心极限定理。

The feasible region for consecutive patterns of permutations is a cycle polytope

We study proportions of consecutive occurrences of permutations of a given size. Specifically, the limit of such proportions on large permutations forms a region, called \emph{feasible region}. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called \emph{overlap graph}. This allows us to compute the dimension, vertices and faces of the polytope, and to determine the equations that define it. Finally we prove that the limit of classical occurrences and consecutive occurrences are in some sense independent. As a consequence, the scaling limi…

Expected value of letters of permutations with a given number of $k$-cycles

In this paper, we study permutations $π\in S_n$ with exactly $m$ transpositions. In particular, we are interested in the expected value of $π(1)$ when such permutations are chosen uniformly at random. When $n$ is even, this expected value is approximated closely by $(n+1)/2$, with an error term that is related to the number isometries of the $(n/2-m)$-dimensional hypercube that move every face. Furthermore, when $k \mid n$, this construction generalizes to allow us to compute the expected value of $π(1)$ for permutations with exactly $m$ $k$-cycles. In this case, the expected value has an erro…

按循环类型划分的置换统计量的矩

Moments of permutation statistics by cycle type

该研究引入正则统计量，通过将其分解为部分置换指示函数线性组合再求矩的两步法，推广了置换统计量矩的相关结论，并证明其大数定律与方差性质，同时给出不依赖表示论的自包含证明。

The feasible region for consecutive patterns of permutations is a cycle polytope

We study proportions of consecutive occurrences of permutations of a given size. Specifically, the feasible limits of such proportions on large permutations form a region, called feasible region. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called overlap graph. This allows us to compute the dimension, vertices and faces of the polytope. Finally, we prove that the limits of classical occurrences and consecutive occurrences are independent, in some sense made precise in the extended abstract. As a consequence, the scaling limit of a sequ…

具有限制圈长的随机置换的圈

Cycles of random permutations with restricted cycle lengths

本文证明了分布共轭不变的随机置换的给定长度的圈数分布的渐近性的一些一般结果。这些结果首先建立应用在即将发表的论文（周期的自由字在几个随机排列与限制周期长度），在那里我们证明结果的周期随机排列，可以写为自由字在几个独立的随机排列。然而，我们也适用于他们在这里证明渐近结果的随机置换的限制周期长度。更具体地说，对于$A$是一组正整数，我们考虑在$\{1,..., n\}$的排列中均匀选择一个随机排列，这些排列的所有周期长度都在$A$中，然后让$n$趋于无穷大。本文改进了Yakymiv（Random A-Permutations：Convergence to a泊松过程）的一个结果，证明了在$A$的一般假设下，该随机置换的定长圈数是渐近独立的，且服从泊松分布.在$A$是有限的情况下，我们证明了这些随机变量的行为是完全不同的：周期长度为$\max A$是占主导地位的。

The number of cycles in random permutations without long cycles is asymptotically Gaussian

For uniform random permutations conditioned to have no long cycles, we prove that the total number of cycles satisfies a central limit theorem. Under additional assumptions on the asymptotic behavior of the set of allowed cycle lengths, we derive asymptotic expansions for the corresponding expected value and variance.

A central limit theorem for cycles of Mallows permutations

Fix $q\neq 1$, and sample $w\in S_n$ from the Mallows measure. We study the distribution of $C_i(w)$, the number of $i$-cycles, as $n$ grows large. When $q<1$, they are jointly Gaussian, and this more or less follows from known ideas, but the regime $q>1$ behaves quite differently. In particular, we show that the even cycles $C_{2i}(w)$ have a mean and variance of order $n$, and jointly converge to Gaussian random variables, while the odd cycles $C_{2i+1}(w)$ have a bounded mean and variance, and converge to $C_{2i+1}(w^{even})$ or $C_{2i+1}(w^{odd})$ for some explicit random permutations $w^{…

The number of cycles of specified normalized length in permutations

We compute the limiting distribution, as n approaches infinity, of the number of cycles of length between gamma n and delta n in a permutation of [n] chosen uniformly at random, for constants gamma, delta such that 1/(k+1) <= gamma < delta <= 1/k for some integer k. This distribution is supported on {0, 1, ... k} and has 0th, 1st, ..., kth moments equal to those of a Poisson distribution with parameter log (delta/gamma). For more general choices of gamma, delta we show that such a limiting distribution exists, which can be given explicitly in terms of certain integrals over intersections of hy…

The Cycle Structure of Permutations Without Long Cycles

We consider the cycle structure of a random permutation $σ$ chosen uniformly from the symmetric group, subject to the constraint that $σ$ does not contain cycles of length exceeding $r.$ We prove that under suitable conditions the distribution of the cycle counts is approximately Poisson and obtain an upper bound on the total variation distance between the distributions using Stein's method of exchangeable pairs. Our results extend the recent work of Betz, Schäfer, and Zeindler.

Long cycle of random permutations with polynomially growing cycle weights

We study the asymptotic behavior of the long cycles of a random permutation of $n$ objects with respect to multiplicative measures with polynomial growing cycle weights. We show that the longest cycle and the length differences between the longest cycles converge, after suitable normalisation, in distribution to iid random variables $(Z_j)_{j\in\mathbb{N}}$ such that $\exp(-Z_j)$ is exponentially distributed. Our method is based on generating functions and the saddle point method.

On the number of cycles in a random permutation

We show that the number of cycles in a random permutation chosen according to generalized Ewens measure is normally distributed and compute asymptotic estimates for the mean and variance.

带循环权重的随机排列的短循环：点过程方法

Short cycles of random permutations with cycle weights: point processes approach

本文用点过程方法研究带循环权重的随机排列短循环的渐近行为，证明其依分布收敛于泊松点过程，并给出循环统计量的极限定理。

Mallows排列模型的$L^1$距离下的循环结构

Cycle structure of Mallows permutation model with the $L^1$ distance

本文研究Mallows排列模型在$L^1$距离下的循环结构，证明循环长度与直径的阶，并揭示小$β$时循环长度分布收敛于Poisson-Dirichlet定律。

Permutation graphs, fast forward permutations, and sampling the cycle structure of a permutation

A permutation P on {1,..,N} is a_fast_forward_permutation_ if for each m the computational complexity of evaluating P^m(x)$ is small independently of m and x. Naor and Reingold constructed fast forward pseudorandom cycluses and involutions. By studying the evolution of permutation graphs, we prove that the number of queries needed to distinguish a random cyclus from a random permutation on {1,..,N} is Theta(N) if one does not use queries of the form P^m(x), but is only Theta(1) if one is allowed to make such queries. We construct fast forward permutations which are indistinguishable from rando…

Fixed points and cycle structure of random permutations

Using the recently developed notion of permutation limits this paper derives the limiting distribution of the number of fixed points and cycle structure for any convergent sequence of random permutations, under mild regularity conditions. In particular this covers random permutations generated from Mallows Model with Kendall's Tau, $μ$ random permutations introduced in [11], as well as a class of exponential families introduced in [15].

关于共轭不变排列的大偏差原理的普适性

About universality of large deviation principles for conjugacy invariant permutations

本文证明了具有少量循环的共轭不变排列的大偏差普适性，并应用于单调子序列长度，涵盖 Ewens 测度。

Random permutations without macroscopic cycles

We consider uniform random permutations of length $n$ conditioned to have no cycle longer than $n^β$ with $0<β<1$, in the limit of large $n$. Since in unconstrained uniform random permutations most of the indices are in cycles of macroscopic length, this is a singular conditioning in the limit. Nevertheless, we obtain a fairly complete picture about the cycle number distribution at various lengths. Depending on the scale at which cycle numbers are studied, our results include Poisson convergence, a central limit theorem, a shape theorem and two different functional central limit theorems.

Cycle type of random permutations: A toolkit

We prove a number of results, new and old, about the cycle type of a random permutation on S_n. Underlying our analysis is the idea that the number of cycles of size k is roughly Poisson distributed with parameter 1/k. In particular, we establish strong results about the distribution of the number of cycles whose lengths lie in a fixed but arbitrary set I. Our techniques are motivated by the theory of sieves in number theory.

Precise asymptotics of longest cycles in random permutations without macroscopic cycles

We consider Ewens random permutations of length $n$ conditioned to have no cycle longer than $n^β$ with $0<β<1$ and to study the asymptotic behaviour as $n\to\infty$. We obtain very precise information on the joint distribution of the lengths of the longest cycles; in particular we prove a functional limit theorem where the cumulative number of long cycles converges to a Poisson process in the suitable scaling. Furthermore, we prove convergence of the total variation distance between joint cycle counts and suitable independent Poisson random variables up to a significantly larger maximal cycle…

Large cycles and a functional central limit theorem for generalized weighted random permutations

The objects of our interest are the so-called $A$-permutations, which are permutations whose cycle length lie in a fixed set $A$. They have been extensively studied with respect to the uniform or the Ewens measure. In this paper, we extend some classical results to a more general weighted probability measure which is a natural extension of the Ewens measure and which in particular allows to consider sets $A_n$ depending on the degree $n$ of the permutation. By means of complex analysis arguments and under reasonable conditions on generating functions we study the asymptotic behaviour of classi…

Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights

We study the model of random permutations of $n$ objects with polynomially growing cycle weights, which was recently considered by Ercolani and Ueltschi, among others. Using saddle-point analysis, we prove that the total variation distance between the process which counts the cycles of size $1, 2, ..., b$ and a process $(Z_1, Z_2, ..., Z_b)$ of independent Poisson random variables converges to $0$ if and only if $b=o(\ell)$ where $\ell$ denotes the length of a typical cycle in this model. By means of this result, we prove a central limit theorem for the order of a permutation and thus extend t…

A generalization of the Erdős-Turán law for the order of random permutation

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on $n$ integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős-Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM$(θ)$ distribution. Our approach is based on using perturbed random walks t…

Spatial random permutations with small cycle weights

We consider the distribution of cycles in two models of random permutations, that are related to one another. In the first model, cycles receive a weight that depends on their length. The second model deals with permutations of points in the space and there is an additional weight that involves the length of permutation jumps. We prove the occurrence of infinite macroscopic cycles above a certain critical density.

分段随机排列

Fragmenting random permutations

Pitman's Saint-Flour讲义中的问题1.5.7：对于每个n，是否存在一个碎片化过程（Π_{n，k}，1 \leq k \leq n），在{1，2，.，，}的分区空间中取值n}，使得k_{n，k}的分布类似于由{1，2，.，n}有条件有k个循环？我们证明答案是肯定的。我们也给出了一般交换Gibbs分拆的部分推广。

Some Connections Between Cycles and Permutations that Fix a Set and Touchard Polynomials and Covers of Multisets

We present a new proof of a fundamental result concerning cycles of random permutations which gives some intuition for the connection between Touchard polynomials and the Poisson distribution. We also introduce a rather novel permutation statistic and study its distribution. This quantity, indexed by $m$, is the number of sets of size $m$ fixed by the permutation. This leads to a new and simpler derivation of the exponential generating function for the number of covers of certain multisets.

Large cycles in random permutations related to the Heisenberg model

We study the weighted version of the interchange process where a permutation receives weight $θ^{\#\mathrm{cycles}}$. For $θ=2$ this is Tóth's representation of the quantum Heisenberg ferromagnet on the complete graph. We prove, for $θ>1$, that large cycles appear at `low temperature'.

The order of large random permutations with cycle weights

The order $O_n(σ)$ of a permutation $σ$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $σ$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdös and Turán who proved in 1965 that $\log O_n$ satisfies a central limit theorem. We extend this result to the so-called \textit{generalized Ewens measure} in a previous paper. In this paper, we establish a local limit theorem as well as, under some extra moment condition, a precise large deviations estimate. These properties are new even for the uniform measure. Furt…

The cycle structure of compositions of random involutions

In this article we consider the cycle structure of compositions of pairs of involutions in the symmetric group S_n chosen uniformly at random. These can be modeled as modified 2-regular graphs, giving rise to exponential generating functions. A composition of two random involutions in S_n typically has about n^(1/2) cycles, and the cycles are characteristically of length n^(1/2). Compositions of two random fixed-point-free involutions, on the other hand, typically have about log n cycles and are closely related to permutations with all cycle lengths even. The number of factorizations of a rand…

Cycles in Mallows random permutations

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…

On averages of randomized class functions on the symmetric groups and their asymptotics

The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices (n points). In this paper, we generalize many aspects of this situation. We introduce random shifts of the eigenvalues of the permutation matrices, in two different ways: independently or not for each subset of eigenvalues associated to the same cycle. We also consider vastly more general functions than the characteristic polynomial of a permutation matrix, by first finding an equivalent definition in terms of cycle-type of the permutation. We consider other…

Asymptotic statistics of cycles in surrogate-spatial permutations

We propose an extension of the Ewens measure on permutations by choosing the cycle weights to be asymptotically proportional to the degree of the symmetric group. This model is primarily motivated by a natural approximation to the so-called spatial random permutations recently studied by V. Betz and D. Ueltschi (hence the name "surrogate-spatial"), but it is of substantial interest in its own right. We show that under the suitable (thermodynamic) limit both measures have the similar critical behaviour of the cycle statistics characterized by the emergence of infinitely long cycles. Moreover, u…

Spatial random permutations and Poisson-Dirichlet law of cycle lengths

We study spatial permutations with cycle weights that are bounded or slowly diverging. We show that a phase transition occurs at an explicit critical density. The long cycles are macroscopic and their cycle lengths satisfy a Poisson-Dirichlet law.

Random permutations with logarithmic cycle weights

We consider random permutations on $\Sn$ with logarithmic growing cycles weights and study asymptotic behavior as the length $n$ tends to infinity. We show that the cycle count process converges to a vector of independent Poisson variables and also compute the total variation distance between both processes. Next, we prove a central limit theorem for the total number of cycles. Furthermore we establish a shape theorem and a functional central limit theorem for the Young diagrams associated to random permutations under this measure. We prove these results using tools from complex analysis and c…

Random Permutations and Queues

Given a growth rule which sequentially constructs random permutations of increasing degree, the stochastic process version of the rencontre problem asks what is the limiting proportion of time that the permutation has no fixed points (singleton cycles). We show that the discrete-time Chinese Restaurant Process (CRP) does not exhibit this limit. We then consider the related embedding of the CRP in continuous time and thereby show that it does have this and other limits of the time averages. By this embedding the cycle structure of the permutation can be represented as a tandem of infinite-serve…

Cycles of free words in several independent random permutations with restricted cycle lengths

In this text, we consider random permutations which can be written as free words in several independent random permutations: firstly, we fix a non trivial word $w$ in letters $g_1,g_1^{-1},..., g_k,g_k^{-1}$, secondly, for all $n$, we introduce a $k$-tuple $s_1(n),..., s_k(n)$ of independent random permutations of $\{1,..., n\}$, and the random permutation $σ_n$ we are going to consider is the one obtained by replacing each letter $g_i$ in $w$ by $s_i(n)$. For example, for $w=g_1g_2g_3g_2^{-1}$, $σ_n=s_1(n)\circ s_2(n)\circ s_3(n)\circ s_2(n)^{-1}$. Moreover, we restrict the set of possible le…

The limit shape of random permutations with polynomially growing cycle weights

In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set $\{1,\dots,n\}$ under a particular class of multiplicative measures. Our method is based on generating functions and complex analysis (saddle point method). We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also compare our approach with the so-called randomization of the cycle counts of permutations and we will study the …

Finite cycle Gibbs measures on permutations of $\mathbb Z^d$

We consider Gibbs distributions on the set of permutations of $\mathbb Z^d$ associated to the Hamiltonian $H(σ):=\sum_{x} V(σ(x)-x)$, where $σ$ is a permutation and $V:\mathbb Z^d\to\mathbb R$ is a strictly convex potential. Call finite-cycle those permutations composed by finite cycles only. We give conditions on $V$ ensuring that for large enough temperature $α>0$ there exists a unique infinite volume ergodic Gibbs measure $μ^α$ concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We constru…

关于蝴蝶排列的最长递增子序列与循环数

On the longest increasing subsequence and number of cycles of butterfly permutations

本文研究由带部分主元高斯消元法生成的随机蝴蝶排列，探讨了其最长递增子序列和循环数问题，建立了幂律界限与中心极限定理，揭示了其与典型随机排列的区别。

群胚基数与随机置换

Groupoid Cardinality and Random Permutations

本文将对称群$S_n$视为概率测度空间，把置换循环数作为随机变量，通过群胚等价性对描述其乘积期望值的循环长度引理进行范畴化，给出了该引理的群胚理论推导依据。

循环置换：度与组合类型

Cyclic Permutations: Degrees and Combinatorial Types

本文枚举对称群中n-循环按度的计数，通过动力系统周期轨道研究共轭类计数，并证明随机n-循环的度分布渐近正态。

多重集合排列的中国餐厅过程

A Chinese restaurant process for multiset permutations

本文提出了一种中国餐厅过程，用于生成多重集合排列，展示了在不同θ值下循环数的分布及中心极限定理。

非对称环面上的有向空间置换

Directed Spatial Permutations on Asymmetric Tori

本文研究二维非对称环面上的随机空间置换，利用空间偏置、动态重抽样与 Schramm 方法证明大循环渐近服从参数为一的 Poisson–Dirichlet 分布。

一维空间随机排列的指数定位

Exponential Localization of Spatial Random Permutations in One Dimension

该研究探讨了一维空间中随机排列的局部化特性，证明了循环在特定尺度上的指数衰减行为，并推测其与随机带状矩阵本征函数的相似性。

关于一个随机循环模型

On a model of random cycles

我们考虑一个模型的随机置换的网站的立方晶格。排列经过加权，以便最好将站点发送到邻居。我们提出了数值证据的过渡到一个阶段的发生与无限的，宏观周期。

The probability of long cycles in interchange processes

We examine the number of cycles of length k in a permutation, as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows to study formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time t, and estimates for the cases of shorter cycles.

Limit distributions for Euclidean random permutations

We study the length of cycles in the model of spatial random permutations in Euclidean space. In this model, for given length $L$, density $ρ$, dimension $d$ and jump density $φ$, one samples $ρL^d$ particles in a $d$-dimensional torus of side length $L$, and a permutation $π$ of the particles, with probability density proportional to the product of values of $φ$ at the differences between a particle and its image under $π$. The distribution may be further weighted by a factor of $θ$ to the number of cycles in $π$. Following Matsubara and Feynman, the emergence of macroscopic cycles in $π$ at …