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Register a new userRandom truncations of Haar distributed matrices and bridges
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Catherine Donati-Martin
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Let $U$ be a Haar distributed matrix in $mathbb U(n)$ or $mathbb O (n)$. In a previous paper, we proved that after centering, the two-parameter process [T^{(n)} (s,t) = sum_{i leq lfloor ns rfloor, j leq lfloor ntrfloor} |U_{ij}|^2] converges in distribution to the bivariate tied-down Brownian bridge. In the present paper, we replace the deterministic truncation of $U$ by a random one, where each row (resp. column) is chosen with probability $s$ (resp. $t$) independently. We prove that the corresponding two-parameter process, after centering and normalization by $n^{-1/2}$ converges to a Gaussian process. On the way we meet other interesting convergences.
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Let U be a Haar distributed unitary matrix in U(n)or O(n). We show that after centering the double index process $$ W^{(n)} (s,t) = sum_{i leq lfloor ns rfloor, j leq lfloor ntrfloor} |U_{ij}|^2 $$ converges in distribution to the bivariate tied-down Brownian bridge. The proof relies on the notion of second order freeness.
We consider the empirical eigenvalue distribution of an $mtimes m$ principal submatrix of an $ntimes n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $frac{m}{n}=alpha$, the empirical spectral measure is well-approximated by a deterministic measure $mu_alpha$ supported on the unit disc. In earlier work, we showed that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $mu_alpha$ is typically of order $sqrt{frac{log(m)}{m}}$ or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.
We consider the empirical eigenvalue distribution of an $mtimes m$ principle submatrix of an $ntimes n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and Reffy identified the limiting spectral measure if $frac{m}{n}toalpha$, as $ntoinfty$; under suitable scaling, the family ${mu_alpha}_{alphain(0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $alpha$) and uniform measure on the unit circle (as $alphato1$). In this note, we prove an explicit concentration inequality which shows that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $mu_alpha$ is typically of order $sqrt{frac{log(m)}{m}}$ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new Coulomb transport inequality due to Chafai, Hardy, and Maida.
Let $O(2n+ell)$ be the group of orthogonal matrices of size $left(2n+ellright)times left(2n+ellright)$ equipped with the probability distribution given by normalized Haar measure. We study the probability begin{equation*} p_{2n}^{left(ellright)} = mathbb{P}left[M_{2n} , mbox{has no real eigenvalues}right], end{equation*} where $M_{2n}$ is the $2ntimes 2n$ left top minor of a $(2n+ell)times(2n+ell)$ orthogonal matrix. We prove that this probability is given in terms of a determinant identity minus a weighted Hankel matrix of size $ntimes n$ that depends on the truncation parameter $ell$. For $ell=1$ the matrix coincides with the Hilbert matrix and we prove begin{equation*} p_{2n}^{left(1right)} sim n^{-3/8}, mbox{ when }n to infty. end{equation*} We also discuss connections of the above to the persistence probability for random Kac polynomials.
Random unitary matrices find a number of applications in quantum information science, and are central to the recently defined boson sampling algorithm for photons in linear optics. We describe an operationally simple method to directly implement Haar random unitary matrices in optical circuits, with no requirement for prior or explicit matrix calculations. Our physically-motivated and compact representation directly maps independent probability density functions for parameters in Haar random unitary matrices, to optical circuit components. We go on to extend the results to the case of random unitaries for qubits.
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