The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles

Abstract

The goal of this paper is to analyse the asymptotic behavior of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables and to establish a central limit theorem for the total number of cycles. Our methods allow us to obtain an asymptotic estimate of the characteristic functions of the different random vectors of interest together with an error estimate, thus having a control on the speed of convergence. In fact we are able to prove a finer convergence for the total number of cycles, namely mod-Poisson convergence. From there we apply previous results on mod-Poisson convergence to obtain Poisson approximation for the total number of cycles as well as large deviations estimates.

Limit Distributions for Euclidean Random Permutations

“…There has been significant activity [11,10,19,6,49,46,36,40,22,34,5,20] in recent years in studying random permutations in which the probability of a permutation is proportional to a product over its cycles of a weight depending on the length of the cycle. The spatial random permutation model has this general form, see Proposition 3.1, but differs from the models studied in this literature in that the weight assigned to a cycle depends both on the length of the cycle and on the size of the permutation.…”

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

“…Subsequently, several properties of permutations have been studied under this measure by many authors and for different classes of parameters, see for instance [17,22,23,24,25,26]. In this paper, the large n statistics of log(O n ) are considered for polynomially growing parameters Θ = (θ m ) m≥1 with θ m = m γ , γ > 0.…”

On the number of cycles in a random permutation

“…This proposition follows immediately from Polya's enumeration theorem with a small calculation. More details can be found for instance in [7,Section 4] Remark. In [7] the characteristic function E exp(itK 0n ) of K 0n was considered.…”