Local Eigenvalue Density for General MANOVA Matrices

We consider random n\times n matrices of the form (XX*+YY*)^{-1/2}YY*(XX*+YY*)^{-1/2}, where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to \log n factors). This result is the analogue of the local Wigner semicircle law and the local Marchenko-Pastur law for general MANOVA matrices.Comment: Several small changes made to the tex

Anomalous diffusion for a class of systems with two conserved quantities

We introduce a class of one dimensional deterministic models of energy-volume conserving interfaces. Numerical simulations show that these dynamics are genuinely super-diffusive. We then modify the dynamics by adding a conservative stochastic noise so that it becomes ergodic. System of conservation laws are derived as hydrodynamic limits of the modified dynamics. Numerical evidence shows these models are still super-diffusive. This is proven rigorously for harmonic potentials

How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise

Classical statistics suggest that for inference purposes one should always use as much data as is available. We study how the presence of market microstructure noise in high-frequency financial data can change that result. We show that the optimal sampling frequency at which to estimate the parameters of a discretely sampled continuous-time model can be finite when the observations are contaminated by market microstructure effects. We then address the question of what to do about the presence of the noise. We show that modelling the noise term explicitly restores the first order statistical effect that sampling as often as possible is optimal. But, more surprisingly, we also demonstrate that this is true even if one misspecifies the assumed distribution of the noise term. Not only is it still optimal to sample as often as possible, but the estimator has the same variance as if the noise distribution had been correctly specified, implying that attempts to incorporate the noise into the analysis cannot do more harm than good. Finally, we study the same questions when the observations are sampled at random time intervals, which are an essential feature of transaction-level data.

Phase space transport in cuspy triaxial potentials: Can they be used to construct self-consistent equilibria?

(Abridged) This paper studies chaotic orbit ensembles evolved in triaxial generalisations of the Dehnen potential which have been proposed to model ellipticals with a strong density cusp that manifest significant deviations from axisymmetry. Allowance is made for a possible supermassive black hole, as well as low amplitude friction, noise, and periodic driving which can mimic irregularities associated with discreteness effects and/or an external environment. The degree of chaos is quantified by determining how (1) the relative number of chaotic orbits and (2) the size of the largest Lyapunov exponent depend on the steepness of the cusp and the black hole mass, and (3) the extent to which Arnold webs significantly impede phase space transport, both with and without perturbations. In the absence of irregularities, chaotic orbits tend to be extremely `sticky,' so that different pieces of the same chaotic orbit can behave very differently for 10000 dynamical times or longer, but even very low amplitude perturbations can prove efficient in erasing many -- albeit not all -- these differences. The implications thereof are discussed both for the structure and evolution of real galaxies and for the possibility of constructing approximate near-equilibrium models using Schwarzschild's method. Much of the observed qualitative behaviour can be reproduced with a toy potential given as the sum of an anisotropic harmonic oscillator and a spherical Plummer potential, which suggests that the results may be generic.Comment: 18 pages, including 19 figures; Accepted for publication by MNRAS; higher quality figures available from http://www.astro.ufl.edu/~siopis/papers

On the group of alternating colored permutations

The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length function with respect to that presentation. Then, we present this group as a covering of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ and use this point of view to give another expression for the length function. We also use this covering to lift several known parameters of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ to the group of alternating colored permutations.Comment: 29 pages, one figure; submitte