2,484 research outputs found
Quantum invariant families of matrices in free probability
We consider (self-adjoint) families of infinite matrices of noncommutative
random variables such that the joint distribution of their entries is invariant
under conjugation by a free quantum group. For the free orthogonal and
hyperoctahedral groups, we obtain complete characterizations of the invariant
families in terms of an operator-valued -cyclicity condition. This is a
surprising contrast with the Aldous-Hoover characterization of jointly
exchangeable arrays.Comment: 33 page
The uniform distributions puzzle
Ordering infinite utility streams: maximal anonymity
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
Cycles of free words in several independent random permutations with restricted cycle lengths
In this text, we consider random permutations which can be written as free
words in several independent random permutations: firstly, we fix a non trivial
word in letters , secondly, for all , we
introduce a -tuple of independent random permutations
of , and the random permutation we are going to
consider is the one obtained by replacing each letter in by .
For example, for , . Moreover, we restrict the set of possible lengths of
the cycles of the 's: we fix sets of positive integers
and suppose that for all , for all , is uniformly distributed on
the set of permutations of which have all their cycle lengths in
. For all positive integer , we are going to give asymptotics, as
goes to infinity, on the number of cycles of length of
. We shall also consider the joint distribution of the random vectors
. We first prove that the order of in a
certain quotient of the free group with generators determines
the rate of growth of the random variables as goes to
infinity. We also prove that in many cases, the distribution of
converges to a Poisson law with parameter and that the random variables
are asymptotically independent. We notice
the surprising fact that from this point of view, many things happen as if
were uniformly distributed on the -th symmetric group.Comment: 28 page
- …