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.
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.Comment: 15 pages, no figure
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 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 "surrogatespatial"), 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 distribution of ordered cycles to the occurrence of a single giant cycle. Our techniques are based on the asymptotic analysis of the corresponding generating functions using Pólya's Enumeration Theorem and complex variable methods.
Abstract. 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 (Z1, Z2, ..., Z b ) of independent Poisson random variables converges to 0 if and only if b = o(ℓ) where ℓ 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 the Erdős-Turán Law to this measure. Furthermore, we prove a Brownian motion limit theorem for the small cycles.
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 groups than the symmetric group, for instance the alternating group and other Weyl groups. Finally, we compute some asymptotics results when n tends to infinity. This last result requires additional ideas: it exploits properties of the Feller coupling, which gives asymptotics for the lengths of cycles in permutations of many points.
The mollification ζ(s) + ζ (s) put forward by Feng is computed by analytic methods coming from the techniques of the ratios conjectures of L-functions. The current situation regarding the percentage of non-trivial zeros of the Riemann zeta-function on the critical line is then clarified.
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