Correlations for the Dyson Brownian motion model with Poisson initial conditions

The circular Dyson Brownian motion model refers to the stochastic dynamics of the log-gas on a circle. It also specifies the eigenvalues of certain parameter-dependent ensembles of unitary random matrices. This model is considered with the initial condition that the particles are non-interacting (Poisson statistics). Jack polynomial theory is used to derive a simple exact expression for the density-density correlation with the position of one particle specified in the initial state, and the position of one particle specified at time $\tau$, valid for all $\beta > 0$. The same correlation with two particles specified in the initial state is also derived exactly, and some special cases of the theoretical correlations are illustrated by comparison with the empirical correlations calculated from the eigenvalues of certain parameter-dependent Gaussian random matrices. Application to fluctuation formulas for time displaced linear statistics in made.Comment: 17 pgs., 2 postscript fig

Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges

For the orthogonal-unitary and symplectic-unitary transitions in random matrix theory, the general parameter dependent distribution between two sets of eigenvalues with two different parameter values can be expressed as a quaternion determinant. For the parameter dependent Gaussian and Laguerre ensembles the matrix elements of the determinant are expressed in terms of corresponding skew-orthogonal polynomials, and their limiting value for infinite matrix dimension are computed in the vicinity of the soft and hard edges respectively. A connection formula relating the distributions at the hard and soft edge is obtained, and a universal asymptotic behaviour of the two point correlation is identified.Comment: 37 pgs., 1fi

Beginner Modeling Exercises

The goal of this paper written as part of the MIT Systems Dynamics in Education Project is to teach the reader how to distinguish between stocks and flows. A stock is an accumulation that is changed over time by inflows and outflows. The reader will gain intuition about stocks and flow through and extensive list of different examples and will practice modeling simple systems with constant flows. STELLA modeling examples include, but are not restricted to, skunks populations, landfills, a bank account and nuclear weapons. Educational levels: High school, Middle school, Undergraduate lower division, Undergraduate upper division

WINGS-CF Face-to-Face Meeting 2004

This report focuses mainly on workshop discussions and has been written from detailed notes taken by workshop scribes. Where there was overlap in discussion topics, some points have been combined: this is not just a transcript of the workshop discussions. The report starts with a summary of the implications for WINGS-CF from the meeting, and an overview of the workshops. For anyone who wants to delve more deeply into how a topic was explored at the gathering, Section 4 gives details of discussion, drawn from notes taken by each group and the "post-it" thoughts provided by participants before the working groups started the discussions

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