Asymptotic behavior of some statistics in Ewens random permutations
Valentin Féray
备注：
32 pages: final version for EJP, produced by the author. An extended abstract of 12 pages, published in the proceedings of AofA 2012, is also available as version 3
发表信息：
Electronic Journal of Probability 18, 76 (2013) 1-32
The purpose of this article is to present a general method to find limiting laws for some renormalized statistics on random permutations. The model considered here is Ewens sampling model, which generalizes uniform random permutations. We describe the asymptotic behavior of a large family of statistics, including the number of occurrences of any given dashed pattern. Our approach is based on the method of moments and relies on the following intuition: two events involving the images of different integers are almost independent.
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…
We consider a generalization of the Ewens measure for the symmetric group, calculating moments of the characteristic polynomial and similar multiplicative statistics. In addition, we study the asymptotic behavior of linear statistics (such as the trace of a permutation matrix or of a wreath product) under this new measure.
We study the number of random permutations needed to invariably generate the symmetric group, $S_n$, when the distribution of cycle counts has the strong $α$-logarithmic property. The canonical example is the Ewens sampling formula, for which the number of $k$-cycles relates to a conditioned Poisson random variable with mean $α/k$. The special case $α=1$ corresponds to uniformly random permutations, for which it was recently shown that exactly four are needed. For strong $α$-logarithmic measures, and almost every $α$, we show that precisely $\left\lceil ( 1- α\log 2 )^{-1} \right\rceil$ permut…
The Ewens sampling formula with parameter $α$ is the distribution on $S_n$ which gives each $π\in S_n$ weight proportional to $α^{C(π)}$, where $C(π)$ is the number of cycles of $π$. We show that, for any fixed $α$, two Ewens-random permutations generate at least $A_n$ with high probability. More generally we work out how many permutations are needed for $α$ growing with $n$. Roughly speaking, two are needed for $0 \leq α\ll n^{1/2}$, three for $n^{1/2} \ll α\ll n^{2/3}$, etc.
The variance of a linear statistic defined on the symmetric group endowed with the Ewens probability is examined. Despite the dependence of the summands, it can be bounded from above by a constant multiple of the sum of variances. We find the exact value of this constant. The analysis of the appearing quadratic forms and eigenvalue search is built upon the exponential matrices and discrete Hahn's polynomials.
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.
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…
It is known from the work of Baik, Deift, and Johansson [1999] that we have Tracy-Widom fluctuations for the longest increasing subsequence of uniform permutations. In this paper, we prove that this result holds also in the case of the Ewens distribution and more generally for a class of random permutation with distribution invariant under conjugation. Moreover, we obtain the convergence of the first components of the associated Young tableaux to the Airy Ensemble as well as the global convergence to the Vershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove that the limiting…
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…
Let the term $k$-representation refer to the permutation representations of the symmetric group $\mathfrak{S}_n$ on $k$-tuples and $k$-subsets as well as the $S^{(n-k,1^k)}$ irreducible representation of $\mathfrak{S}_n$. Endow $\mathfrak{S}_n$ with the Ewens distribution and let $α$ and $β$ be linearly independent irrational numbers over $\mathbb{Q}$. Then for fixed $k > 1$ we show that as $n \to \infty$, the normalized count of the number of eigenangles in a fixed interval $(α, β)$ of a $k$-representation evaluated at a random element $σ\in \mathfrak{S}_n$ converges weakly to a compactly sup…
Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting non-universality phenomenon. Though they have bounded variance, their fluctuations are asymptotically non-Gaussian but infinitely divisible. The fluctuations are asymptotically Gaussian for less smooth linear statistics for which the variance diverges. The degree of smoothness is measured in terms of the quality of the trapezoidal approximations of the integral of the observable.
We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough test function $f$ to all the determinations of the eigenangles of the permutations, we get a convergence in distribution when the order of the permutation tends to infinity. Two distinct kinds of limit appear: if $f(0)\neq 0$, we have a central limit theorem with a logarithmic variance, and if $f(0) = 0$, the convergence holds without normalization and the li…
We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation group. We also look at some modifications of permutation matrices where the entries equal to one are replaced by i.i.d uniform variables on the unit circle. Once appropriately normalized and scaled, we show that the characteristic polynomial converges in distribution on every compact subset of $\mathbb{C}$ to an explicit limiting entire function, when the si…
Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the two-parameter Poisson-Dirichlet distribution. A sample of size n is selected from the population. The total number of different types and the number of types appearing in the sample with a fixed frequency are important statistics. In this paper we establish the moderate deviation principles for these quantities. The corresponding rate functions are explicitly identified, which help revealing a critical scale and understanding the exact role of the parameters. Conditional, …
Ewens-Pitman model has been successfully applied to various fields including Bayesian statistics. There are four important estimators $K_{n},M_{l,n}$,$K_{m}^{(n)},M_{l,m}^{(n)}$. In particular, $M_{1,n}, M_{1,m}^{(n)}$ are related to discovery probability. Their asymptotic behavior, such as large deviation principle, has already been discussed in [4],[1] and [2]. Moderate deviation principle is also discussed in [3] with some speed restriction. In this article, we will apply complex asymptotic analysis to show that this speed restriction is unnecessary.
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…
We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show that the behavior at different points is independent in the limit and are asymptotically normal. Our methods enables us to study more general matrices, closely related to permutation matrices, and multiplicative class functions.
Equip the symmetric group $\mathfrak{S}_n$ with the Ewens distribution. We study the eigenvalue point process of the permutation representation of $\mathfrak{S}_n$ on $k$-tuples of distinct integers chosen from the set $\{1,2,...,n\}$. Taking $n \to \infty$, we find the limiting point process in the microscopic regime, i.e. when the eigenvalue point process is viewed at the scale of the mean eigenvalue spacing. A formula for the limiting eigenvalue gap probability in an interval is also given. In certain cases, a power series representation exists and a combinatorial procedure is given for com…
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 are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens' distribution of a given parameter $θ>0$, and its modification where entries equal to $1$ in the matrices are replaced by independent random variables uniformly distributed on the unit circle. For the elements of each ensemble, we focus on the random numbers of eigenvalues lying in some specified arcs of the unit circle. We show that for a finite number of fixed arcs, the fluctuation of the numbers of eigenvalues belonging to them is asymptotically Gaussian. Moreover, …
We consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle. For each of these two ensembles of matrices, rescaling properly the eigenangles provides a limiting point process as the size of the matrices goes to infinity. If $J$ is an interval of $\mathbb{R}$, we show that, as the length of $J$ tends to infinity, the number of points lying in $J$ of the limiting point process related…
Following Talagrand's concentration results for permutations picked uniformly at random from a symmetric group [Tal95], Luczak and McDiarmid have generalized it to more general groups G of permutations which act suitably 'locally'. Here we extend their results by setting transport-entropy inequalities on these permutations groups. Talagrand and Luczak-Mc-Diarmid concentra- tion properties are consequences of these inequalities. The results are also gen- eralised to a larger class of measures including Ewens distributions of arbitrary parameter $θ$ on the symmetric group. By projection, we deri…
In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter $θ>0$) by replacing the entries equal to one by more general non-vanishing complex random variables. For these ensembles, in contrast with more classical models as the Gaussian Unitary Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very explicitly computed by using the cycle structure of the permutations. Moreover, by using the so-called virtual permutations, first introduced by Kerov, Olsha…
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…
Consider a random permutation of $\{1, \ldots, \lfloor n^{t_2}\rfloor\}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n, t)$ denote the number of its cycles, where $t\equiv (t_1, t_2)\in\mathbb [0, 1]^2$. Next, consider a sample drawn from a large, neutral population of haploid individuals subject to mutation under the infinitely many sites model of Kimura whose genealogy is governed by Kingman's coalescent. Let $S(n, t)$ count the number of segregating sites in a sample of size $\lfloor n^{t_2}\rfloor$ when mutations arrive at rate $t_1/2$. Our main result comprises tw…
Feller (1945) provided a coupling between the counts of cycles of various sizes in a uniform random permutation of $[n]$ and the spacings between successes in a sequence of $n$ independent Bernoulli trials with success probability $1/n$ at the $n$th trial. Arratia, Barbour and Tavaré (1992) extended Feller's coupling, to associate cycles of random permutations governed by the Ewens $(θ)$ distribution with spacings derived from independent Bernoulli trials with success probability $θ/(n-1+θ)$ at the $n$th trial, and to conclude that in an infinite sequence of such trials, the numbers of spacing…
Type-B permutation tableaux are combinatorial objects introduced by Lam and Williams that have an interesting connection with the partially asymmetric simple exclusion process (PASEP). In this paper, we compute the expected value of several statistics on these tableaux. Some of these computations are motivated by a similar paper on permutation tableaux. Others are motivated by the PASEP. In particular, we compute the expected number of rows, unrestricted rows, diagonal ones, adjacent south steps, and adjacent west steps.
Consider a random permutation of $\{1, \ldots, \lfloor n^{t_2}\rfloor\}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n, t)$ denote the number of its cycles, where $t\equiv (t_1, t_2)\in\mathbb [0, 1]^2$. Next, consider a sample drawn from a large, neutral population of haploid individuals subject to mutation under the infinitely many sites model of Kimura whose genealogy is governed by Kingman's coalescent. Let $S(n, t)$ count the number of segregating sites in a sample of size $\lfloor n^{t_2}\rfloor$ when mutations arrive at rate $t_1/2$. We show that $K(n, (t_1/\log…
We study (asymmetric) $U$-statistics based on a stationary sequence of $m$-dependent variables; moreover, we consider constrained $U$-statistics, where the defining multiple sum only includes terms satisfying some restrictions on the gaps between indices. Results include a law of large numbers and a central limit theorem. Special attention is paid to degenerate cases where, after the standard normalization, the asymptotic variance vanishes; in these cases non-normal limits occur after a different normalization. The results are motivated by applications to pattern matching in random strings and…
We study statistical properties of the random variables $X_σ(π)$, the number of occurrences of the pattern $σ$ in the permutation $π$. We present two contrasting approaches to this problem: traditional probability theory and the ``less traditional'' computational approach. Through the perspective of the first one, we prove that for any pair of patterns $σ$ and $τ$, the random variables $X_σ$ and $X_τ$ are jointly asymptotically normal (when the permutation is chosen from $S_{n}$). From the other perspective, we develop algorithms that can show asymptotic normality and joint asymptotic normalit…
This paper develops techniques to study the number of descents in random permutations via martingales. We relax an assumption in the Berry-Esseen theorem of Bolthausen (1982) to extend the theorem's scope to martingale differences of time-dependent variances. This extension leads to a new proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order $1/\sqrt{n}.$ The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics inclu…
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School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）；Department of Statistics and Data Science, Shenzhen International Center for Mathematics, Southern University of Science and Technology（南方科技大学深圳国际数学中心统计与数据科学系）；Department of Statistics and Data Science, Southern University of Science and Technology（南方科技大学统计与数据科学系）
