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Michael Anshelevich

  1. Quantum Free Yang-Mills on the Plane.

    Пнд, 06/13/2011 - 15:01 — SomNambul
    Authors: Michael Anshelevich, Ambar N. Sengupta
    Subjects: Operator Algebras
    Abstract

    We construct a free-probability quantum Yang-Mills theory on the two
    dimensional plane, determine the Wilson loop expectation values, and show that
    this theory is the $N=\infty$ limit of U(N) quantum Yang-Mills theory on the
    plane.

  2. Two-state free Brownian motions.

    Втр, 06/08/2010 - 15:01 — SomNambul
    Authors: Michael Anshelevich
    Subjects: Operator Algebras
    Abstract

    In a two-state free probability space $(A, \phi, \psi)$, we define an
    algebraic two-state free Brownian motion to be a process with two-state freely
    independent increments whose two-state free cumulant generating function is
    quadratic. Note that a priori, the distribution of the process with respect to
    the second state $\psi$ is arbitrary. We show, however, that if $A$ is a von
    Neumann algebra, the states $\phi, \psi$ are normal, and $\phi$ is faithful,
    then there is only a one-parameter family of such processes.

  3. The free Meixner class for pairs of measures.

    Втр, 03/23/2010 - 16:01 — SomNambul
    Authors: Michael Anshelevich, Wojciech Młotkowski
    Subjects: Operator Algebras
    Abstract

    We investigate in more detail the two-state free convolution semigroups of
    pairs of measures whose Jacobi parameters are linear in the convolution
    parameter $t$. These semigroups were constructed in arXiv:1001.1540, where we
    also showed that measures with the analogous property for the usual and free
    convolution are exactly the classical, resp. free Meixner classes. The class of
    measures in this paper has not been considered explicitly before, but we show
    that it also has Meixner-type properties.

  4. Free Infinite Divisibility for Q-Gaussians.

    Пт, 03/05/2010 - 16:01 — SomNambul
    Authors: Michael Anshelevich, Marek Bozejko, Franz Lehner, Serban Teodor Belinschi
    Subjects: Operator Algebras
    Abstract

    We prove that the q-Gaussian distribution introduced by Bozejko and Speicher
    is freely infinitely divisible for all q between zero and one.

  5. Semigroups of distributions with linear Jacobi parameters.

    Втр, 01/12/2010 - 16:01 — SomNambul
    Authors: Michael Anshelevich, Wojciech Młotkowski
    Subjects: Combinatorics
    Abstract

    We show that a convolution semigroup of measures has Jacobi parameters linear
    in the convolution parameter $t$ if and only if the measures come from the
    Meixner class. Moreover, we prove the parallel result, in a more explicit way,
    for the free convolution and the free Meixner class. We then construct the
    class of measures satisfying the same property for the two-state free
    convolution. This class has not been considered explicitly before, but we
    describe its relation to the two-state free Laha-Lukacs characterization, and
    to the $q=0$ case of quadratic harnesses.

  6. Bochner-Pearson-type characterization of the free Meixner class.

    Втр, 09/08/2009 - 12:16 — SomNambul
    Authors: Michael Anshelevich
    Subjects: Combinatorics
    Abstract

    The operator $L_\mu: f \mapsto \int \frac{f(x) - f(y)}{x - y} d\mu(y)$ is,
    for a compactly supported measure $\mu$ with an $L^3$ density, a closed,
    densely defined operator on $L^2(\mu)$. We show that the operator $Q = p
    L_\mu^2 - q L_\mu$ has polynomial eigenfunctions if and only if $\mu$ is a free
    Meixner distribution. The only time $Q$ has orthogonal polynomial
    eigenfunctions is if $\mu$ is a semicircular distribution. More generally, the
    only time the operator $p (L_\nu L_\mu) - q L_\mu$ has orthogonal polynomial
    eigenfunctions is when $\mu$ and $\nu$ are related by a Jacobi shift.

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