214 research outputs found
Asymptotic behavior of some statistics in Ewens random permutations
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.Comment: 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
Central limit theorems for patterns in multiset permutations and set partitions
We use the recently developed method of weighted dependency graphs to prove
central limit theorems for the number of occurrences of any fixed pattern in
multiset permutations and in set partitions. This generalizes results for
patterns of size 2 in both settings, obtained by Canfield, Janson and
Zeilberger and Chern, Diaconis, Kane and Rhoades, respectively.Comment: version 2 (52 pages) implements referee's suggestions and uses
journal layou
Weighted dependency graphs
The theory of dependency graphs is a powerful toolbox to prove asymptotic
normality of sums of random variables. In this article, we introduce a more
general notion of weighted dependency graphs and give normality criteria in
this context. We also provide generic tools to prove that some weighted graph
is a weighted dependency graph for a given family of random variables.
To illustrate the power of the theory, we give applications to the following
objects: uniform random pair partitions, the random graph model ,
uniform random permutations, the symmetric simple exclusion process and
multilinear statistics on Markov chains. The application to random permutations
gives a bivariate extension of a functional central limit theorem of Janson and
Barbour. On Markov chains, we answer positively an open question of Bourdon and
Vall\'ee on the asymptotic normality of subword counts in random texts
generated by a Markovian source.Comment: 57 pages. Third version: minor modifications, after review proces
Cyclic inclusion-exclusion
Following the lead of Stanley and Gessel, we consider a morphism which
associates to an acyclic directed graph (or a poset) a quasi-symmetric
function. The latter is naturally defined as multivariate generating series of
non-decreasing functions on the graph. We describe the kernel of this morphism,
using a simple combinatorial operation that we call cyclic inclusion-exclusion.
Our result also holds for the natural noncommutative analog and for the
commutative and noncommutative restrictions to bipartite graphs. An application
to the theory of Kerov character polynomials is given.Comment: comments welcom
Gaussian fluctuations of Young diagrams and structure constants of Jack characters
In this paper, we consider a deformation of Plancherel measure linked to Jack
polynomials. Our main result is the description of the first and second-order
asymptotics of the bulk of a random Young diagram under this distribution,
which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the
first order asymptotics) and Kerov (for the second order asymptotics). This
gives more evidence of the connection with Gaussian -ensemble, already
suggested by some work of Matsumoto.
Our main tool is a polynomiality result for the structure constant of some
quantities that we call Jack characters, recently introduced by Lassalle. We
believe that this result is also interested in itself and we give several other
applications of it.Comment: 71 pages. Minor modifications from version 1. An extended abstract of
this work, with significantly fewer results and a different title, is
available as arXiv:1201.180
Shifted symmetric functions and multirectangular coordinates of Young diagrams
In this paper, we study shifted Schur functions , as well as a
new family of shifted symmetric functions linked to Kostka
numbers. We prove that both are polynomials in multi-rectangular coordinates,
with nonnegative coefficients when written in terms of falling factorials. We
then propose a conjectural generalization to the Jack setting. This conjecture
is a lifting of Knop and Sahi's positivity result for usual Jack polynomials
and resembles recent conjectures of Lassalle. We prove our conjecture for
one-part partitions.Comment: 2nd version: minor modifications after referee comment
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