2,484 research outputs found

    Quantum invariant families of matrices in free probability

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    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 RR-cyclicity condition. This is a surprising contrast with the Aldous-Hoover characterization of jointly exchangeable arrays.Comment: 33 page

    The uniform distributions puzzle

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    Ordering infinite utility streams: maximal anonymity

    Asymptotic behavior of some statistics in Ewens random permutations

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    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

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    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 ww in letters g1,g1−1,...,gk,gk−1g_1,g_1^{-1},..., g_k,g_k^{-1}, secondly, for all nn, we introduce a kk-tuple s1(n),...,sk(n)s_1(n),..., s_k(n) of independent random permutations of {1,...,n}\{1,..., n\}, and the random permutation σn\sigma_n we are going to consider is the one obtained by replacing each letter gig_i in ww by si(n)s_i(n). For example, for w=g1g2g3g2−1w=g_1g_2g_3g_2^{-1}, σn=s1(n)∘s2(n)∘s3(n)∘s2(n)−1\sigma_n=s_1(n)\circ s_2(n)\circ s_3(n)\circ s_2(n)^{-1}. Moreover, we restrict the set of possible lengths of the cycles of the si(n)s_i(n)'s: we fix sets A1,...,AkA_1,..., A_k of positive integers and suppose that for all nn, for all ii, si(n)s_i(n) is uniformly distributed on the set of permutations of {1,...,n}\{1,..., n\} which have all their cycle lengths in AiA_i. For all positive integer ll, we are going to give asymptotics, as nn goes to infinity, on the number Nl(σn)N_l(\sigma_n) of cycles of length ll of σn\sigma_n. We shall also consider the joint distribution of the random vectors (N1(σn),...,Nl(σn))(N_1(\sigma_n),..., N_l(\sigma_n)). We first prove that the order of ww in a certain quotient of the free group with generators g1,...,gkg_1,..., g_k determines the rate of growth of the random variables Nl(σn)N_l(\sigma_n) as nn goes to infinity. We also prove that in many cases, the distribution of Nl(σn)N_l(\sigma_n) converges to a Poisson law with parameter 1/l1/l and that the random variables N1(σn),N2(σn),...N_1(\sigma_n),N_2(\sigma_n), ... are asymptotically independent. We notice the surprising fact that from this point of view, many things happen as if σn\sigma_n were uniformly distributed on the nn-th symmetric group.Comment: 28 page
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