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Register a new userOperator valued random matrices and asymptotic freeness
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Weihua Liu
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We show that the limit laws of random matrices, whose entries are conditionally independent operator valued random variables having equal second moments proportional to the size of the matrices, are operator valued semicircular laws. Furthermore, we prove an operator valued analogue of Voiculescus asymptotic freeness theorem. By replacing conditional independence with Boolean independence, we show that the limit laws of the random matrices are Operator-valued Bernoulli laws.
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We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives a example where the partial transpose produces freeness at the operator level. Finally we investigate the case of real Wishart matrices.
We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a $C^*$-algebra $mathcal{A}$. We define an $mathcal{A}$-valued chordal Loewner chain as a subordination chain of analytic self-maps of the $mathcal{A}$-valued upper half-plane, such that each $F_t$ is the reciprocal Cauchy transform of an $mathcal{A}$-valued law $mu_t$, such that the mean and variance of $mu_t$ are continuous functions of $t$. We relate $mathcal{A}$-valued Loewner chains to processes with $mathcal{A}$-valued free or monotone independent independent increments just as was done in the scalar case by Bauer (Lowners equation from a non-commutative probability perspective, J. Theoretical Prob., 2004) and Schei{ss}inger (The Chordal Loewner Equation and Monotone Probability Theory, Inf. Dim. Anal., Quantum Probability, and Related Topics, 2017). We show that the Loewner equation $partial_t F_t(z) = DF_t(z)[V_t(z)]$, when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains $F_t$ and vector fields $V_t(z)$ of the form $V_t(z) = -G_{ u_t}(z)$ where $ u_t$ is a generalized $mathcal{A}$-valued law. Based on the Loewner equation, we derive a combinatorial expression for the moments of $mu_t$ in terms of $ u_t$. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws $mu_t$. Finally, we prove a version of the monotone central limit theorem which describes the behavior of $F_t$ as $t to +infty$ when $ u_t$ has uniformly bounded support.
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study their associated convolutions via Voiculescus fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, Boolean and monotone independence convolutions, we will focus on two important convolutions, which are orthogonal and subordination additive convolutions. We show that the operator-valued subordination functions, which come from the free additive convolutions or the operator-valued free convolution powers, are reciprocal Cauchy transforms of operator-valued random variables which are uniquely determined up to Voiculescus fully matricial function theory. In the end, we study relations between certain convolutions and transforms in $C^*$-operator valued probability.
In this paper two independent and unitarily invariant projection matrices P(N) and Q(N) are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size $N$ converges to infinity. The result is formulated on the tracial state space $TS({cal A})$ of the universal $C^*$-algebra ${cal A}$ generated by two selfadjoint projections. The random pair $(P(N),Q(N))$ determines a random tracial state $tau_N in TS({cal A})$ and $tau_N$ satisfies the large deviation. The rate function is in close connection with Voiculescus free entropy defined for pairs of projections.
We prove that any non commutative polynomial of r independent copies of Wigner matrices converges a.s. towards the polynomial of r free semicircular variables in operator norm. This result extends a previous work of Haagerup and Thorbjornsen where GUE matrices are considered, as well as the classical asymptotic freeness for Wigner matrices (i.e. convergence of the moments) proved by Dykema. We also study the Wishart case.
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