1,109 research outputs found
Cycle structure of random permutations with cycle weights
We investigate the typical cycle lengths, the total number of cycles, and the
number of finite cycles in random permutations whose probability involves cycle
weights. Typical cycle lengths and total number of cycles depend strongly on
the parameters, while the distributions of finite cycles are usually
independent Poisson random variables.Comment: 22 pages, 2 figure
Random permutation matrices under the generalized Ewens measure
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.Comment: Published in at http://dx.doi.org/10.1214/12-AAP862 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Functional limit theorems for random regular graphs
Consider d uniformly random permutation matrices on n labels. Consider the
sum of these matrices along with their transposes. The total can be interpreted
as the adjacency matrix of a random regular graph of degree 2d on n vertices.
We consider limit theorems for various combinatorial and analytical properties
of this graph (or the matrix) as n grows to infinity, either when d is kept
fixed or grows slowly with n. In a suitable weak convergence framework, we
prove that the (finite but growing in length) sequences of the number of short
cycles and of cyclically non-backtracking walks converge to distributional
limits. We estimate the total variation distance from the limit using Stein's
method. As an application of these results we derive limits of linear
functionals of the eigenvalues of the adjacency matrix. A key step in this
latter derivation is an extension of the Kahn-Szemer\'edi argument for
estimating the second largest eigenvalue for all values of d and n.Comment: Added Remark 27. 39 pages. To appear in Probability Theory and
Related Field
Random permutations without macroscopic cycles
We consider uniform random permutations of length n conditioned to have no cyclelonger than nβ 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
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