609 research outputs found

### 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

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