Fixed points and cycle structure of random permutations
Sumit Mukherjee
备注：
Minor updates in presentation. The definition of cycles is now corrected, and Theorem 1.4 has been updated to reflect these changes. Electron. J. Probab. 21 (2016), paper no. 40
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].
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.
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.
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.
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…
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…
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…
We present general links between statistics of non-Hermitian random matrices and the distribution of the number of cycles of some specific random permutations. In particular, we derive explicit formulas for the generating functions of the number of cycles in the commutator $[σ,τ] = στσ^{-1} τ^{-1}$ where $σ$ is uniformly distributed, and $τ$ is either one cycle, the product of two cycles of same size, or the product of many transpositions.
We use moment method to understand the cycle structure of the composition of independent invariant permutations. We prove that under a good control on fixed points and cycles of length 2, the limiting joint distribution of the number of small cycles is the same as in the uniform case i.e. for any positive integer k, the number of cycles of length k converges to the Poisson distribution with parameter 1/k and is asymptotically independent of the number of cycles of length k' different from k.
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…
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…
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…
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.
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^{…
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…
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 …
