Tridiagonal realization of the anti-symmetric Gaussian β\beta-ensemble

The Householder reduction of a member of the anti-symmetric Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter $\beta$, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of $\{q_i\}$, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the anti-symmetric tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real anti-symmetric tridiagonal matrices, its eigenvalues and $\{q_i\}$. The third proof maps matrices from the anti-symmetric Gaussian $\beta$-ensemble to those realizing particular examples of the Laguerre $\beta$-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Pr\"ufer phases of the random matrices.Comment: 22 pages; replaced with a new version containing orthogonal transformation proof for both cases (Method III

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

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 $G$, the spherical ensemble of the product of an inverse Ginibre matrix and a Ginibre matrix $G_1^{-1} G_2$, 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 $\beta = 4$, 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

For products $P_N$ of $N$ random matrices of size $d \times d$, there is a natural notion of finite $N$ Lyapunov exponents $\{\mu_i\}_{i=1}^d$. In the case of standard Gaussian random matrices with real, complex or real quaternion elements, and extended to the general variance case for $\mu_1$, methods known for the computation of $\lim_{N \to \infty} \langle \mu_i \rangle$ are used to compute the large $N$ form of the variances of the exponents. Analogous calculations are performed in the case that the matrices making up $P_N$ 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 $P_N$.Comment: 15 page

Exact calculation of the ground state single-particle Green's function for the 1/r21/r^2 quantum many body system at integer coupling

The ground state single particle Green's function describing hole propagation is calculated exactly for the $1/r^2$ quantum many body system at integer coupling. The result is in agreement with a recent conjecture of Haldane.Comment: Late