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We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analog of the Jacobi ensemble:with Reδ > −1/2. If e is a cyclic vector for a unitary n × n matrix U , the spectral measure of the pair (U , e) is well parameterized by its Verblunsky coefficients (α 0 , . . . , α n−1 ). We introduce here a deformation (γ 0 , . . . , γ n−1 ) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r(γ 0 ) · · · r(γ n−1 ) of elementary reflections parameterized by these coefficients. If γ 0 , . . . , γ n−1 are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above.These deformed Verblunsky coefficients also allow us to prove that, in the regime δ = δ(n) with δ(n)/n → βd/2, the spectral measure and the empirical spectral distribution

Homogeneous fragmentations describe the evolution of a unit mass that breaks down randomly into pieces as time passes. They can be thought of as continuous time analogues of a certain type of branching random walk, which suggests the use of time-discretization to shift known results from the theory of branching random walks to the fragmentation setting. In particular, this yields interesting information about the asymptotic behaviour of fragmentations.On the other hand, homogeneous fragmentations can also be investigated using a powerful technique of discretization of space due to Kingman, namely, the theory of exchangeable partitions of N. Spatial discretization is especially well suited to the direct development for continuous times of the conceptual method of probability tilting of Lyons, Pemantle and Peres.

In the theory of orthogonal polynomials, sum rules are remarkable relationships between a functional defined on a subset of all probability measures involving the reverse KullbackLeibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. Killip and Simon (Killip and Simon (2003)) have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution on [−2, 2]. This sum rule includes a contribution of the atomic part of the measure away from [−2, 2]. In this paper, we recover this sum rule by using probabilistic tools on random matrices. Furthermore, we obtain new (up to our knowledge) magic sum rules for the reverse Kullback-Leibler divergence with respect to the Marchenko-Pastur or Kesten-McKay distributions. As in the semicircular case, these formulas include a contribution of the atomic part appearing away from the support of the reference measure.

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We investigated some structural and transport properties of semiconducting ReSi2−δ . In the literature this silicides is reported to crystallize in an orthorhombic structure and to be stoichiometric ReSi2. Our investigations clearly show that the stable composition is ReSi1.75 crystallizing in the space group P1. Transport measurements show thermally activated behavior at high temperatures with one (or two) energy gap Eg=0.16 (0.30 eV). We also report Hall-effect measurements on this material: we found that RH is positive between 30 and 660 K and at room temperature the Hall number nH=1/eRH is equal to 3.7×1018 cm−3. The Hall mobility at room temperature is relatively high (μH=370 cm2/V s) for a single crystal.