609 research outputs found

    Analogies between random matrix ensembles and the one-component plasma in two-dimensions

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    The eigenvalue PDF for some well known classes of non-Hermitian random matrices --- the complex Ginibre ensemble for example --- can be interpreted as the Boltzmann factor for one-component plasma systems in two-dimensional domains. We address this theme in a systematic fashion, identifying the plasma system for the Ginibre ensemble of non-Hermitian Gaussian random matrices GG, the spherical ensemble of the product of an inverse Ginibre matrix and a Ginibre matrix G1βˆ’1G2G_1^{-1} G_2, and the ensemble formed by truncating unitary matrices, as well as for products of such matrices. We do this when each has either real, complex or real quaternion elements. One consequence of this analogy is that the leading form of the eigenvalue density follows as a corollary. Another is that the eigenvalue correlations must obey sum rules known to characterise the plasma system, and this leads us to a exhibit an integral identity satisfied by the two-particle correlation for real quaternion matrices in the neighbourhood of the real axis. Further random matrix ensembles investigated from this viewpoint are self dual non-Hermitian matrices, in which a previous study has related to the one-component plasma system in a disk at inverse temperature Ξ²=4\beta = 4, and the ensemble formed by the single row and column of quaternion elements from a member of the circular symplectic ensemble.Comment: 25 page

    Asymptotics of finite system Lyapunov exponents for some random matrix ensembles

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    For products PNP_N of NN random matrices of size dΓ—dd \times d, there is a natural notion of finite NN Lyapunov exponents {ΞΌi}i=1d\{\mu_i\}_{i=1}^d. In the case of standard Gaussian random matrices with real, complex or real quaternion elements, and extended to the general variance case for ΞΌ1\mu_1, methods known for the computation of lim⁑Nβ†’βˆžβŸ¨ΞΌi⟩\lim_{N \to \infty} \langle \mu_i \rangle are used to compute the large NN form of the variances of the exponents. Analogous calculations are performed in the case that the matrices making up PNP_N are products of sub-blocks of random unitary matrices with Haar measure. Furthermore, we make some remarks relating to the coincidence of the Lyapunov exponents and the stability exponents relating to the eigenvalues of PNP_N.Comment: 15 page
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