Profiles of permutations (original) (raw)

Random permutations with logarithmic cycle weights

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2020

We consider random permutations on S n 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 combinatorics. In particular we have to apply the method of singularity analysis to generating functions of the form exp (− log(1 − z)) k+1 with k ≥ 1, which have not yet been studied in the literature.

On Statistics of Permutations Chosen From the Ewens Distribution

Combinatorics, Probability and Computing, 2014

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial moments, we establish necessary and sufficient conditions for the weak convergence of distributions to discrete laws. More attention is paid to the Poisson limit distribution. The particular case of the number-of-cycles function is analysed in more detail. The results can be applied to statistics defined on random permutation matrices.

On Some Densities in the Set of Permutations

The Electronic Journal of Combinatorics, 2010

The asymptotic density of random permutations with given properties of the kth shortest cycle length is examined. The approach is based upon the saddle point method applied for appropriate sums of independent random variables.

Local probabilities for random permutations without long cycles

We explore the probability that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r. Asymptotic formulas valid in specified regions for the ratio n/r are obtained using the saddle point method combined with ideas originated in analytic number theory. Theorem 1 and its detailed proof are included to rectify formulas for small r which have been announced by a few other authors.

Asymptotic Statistics of Cycles in Surrogate-Spatial Permutations

Communications in Mathematical Physics

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, using a greater analytic tractability of the surrogate-spatial model, we obtain a number of new results about the asymptotic distribution of the cycle lengths (both small and large) in the full range of subcritical, critical and supercritical domains. In particular, in the supercritical regime there is a parametric "phase transition" from the Poisson-Dirichlet limiting distri...

On the frequency of permutations containing a long cycle

Journal of Algebra, 2006

A general explicit upper bound is obtained for the proportion P (n, m) of elements of order dividing m, where n − 1 ≤ m ≤ cn for some constant c, in the finite symmetric group S n. This is used to find lower bounds for the conditional probabilities that an element of S n or A n contains an r-cycle, given that it satisfies an equation of the form x rs = 1 where s ≤ 3. For example, the conditional probability that an element x is an n-cycle, given that x n = 1, is always greater than 2/7, and is greater than 1/2 if n does not divide 24. Our results improve estimates of these conditional probabilities in earlier work of the authors with Beals, Leedham-Green and Seress, and have applications for analysing black-box recognition algorithms for the finite symmetric and alternating groups.

A product of invariant random permutations has the same small cycle structure as uniform

Electronic Communications in Probability, 2020

We use moment method to understand the cycle structure of the composition of two 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 = k.

The Berry–Esseen Bound in the Theory of Random Permutations

1998

The convergence rate in the central limit theorem for linear combinations of the cycle lengths of a random permutation is examined. It is shown that, in contrast to the Berry-Esseen theorem, the optimal estimate in terms of the sum of the third absolute moments has the exponent 2/3.

The random (n-k)-cycle to transpositions walk on the symmetric group

arXiv (Cornell University), 2017

We study the rate of convergence of the Markov chain on Sn which starts with a random (n−k)-cycle for a fixed k ≥ 1, followed by random transpositions. The convergence to the stationary distribution turns out to be of order n. We show that after cn + ln k 2 n steps for c > 0, the law of the Markov chain is close to the uniform distribution. The character of the defining representation is used as test function to obtain a lower bound for the total variation distance. We identify the asymptotic distribution of the test function given the law of the Markov chain for the (n − 1)-cycle case. The upper bound relies on estimates for the difference of normalized characters.