Asymptotics of skew orthogonal polynomials

Exact integral expressions of the skew orthogonal polynomials involved in Orthogonal (beta=1) and Symplectic (beta=4) random matrix ensembles are obtained: the (even rank) skew orthogonal polynomials are average characteristic polynomials of random matrices. From there, asymptotics of the skew orthogonal polynomials are derived.Comment: 17 pages, Late

Solution of a Generalized Stieltjes Problem

We present the exact solution for a set of nonlinear algebraic equations $\frac{1}{z_l}= \pi d + \frac{2 d}{n} \sum_{m \neq l} \frac{1}{z_l-z_m}$. These were encountered by us in a recent study of the low energy spectrum of the Heisenberg ferromagnetic chain \cite{dhar}. These equations are low $d$ (density) ``degenerations'' of more complicated transcendental equation of Bethe's Ansatz for a ferromagnet, but are interesting in themselves. They generalize, through a single parameter, the equations of Stieltjes, $x_l = \sum_{m \neq l} 1/(x_l-x_m)$, familiar from Random Matrix theory. It is shown that the solutions of these set of equations is given by the zeros of generalized associated Laguerre polynomials. These zeros are interesting, since they provide one of the few known cases where the location is along a nontrivial curve in the complex plane that is determined in this work. Using a ``Green's function'' and a saddle point technique we determine the asymptotic distribution of zeros.Comment: 19 pages, 4 figure

Non-Maxwellian Proton Velocity Distributions in Nonradiative Shocks

The Balmer line profiles of nonradiative supernova remnant shocks provide the means to measure the post-shock proton velocity distribution. While most analyses assume a Maxwellian velocity distribution, this is unlikely to be correct. In particular, neutral atoms that pass through the shock and become ionized downstream form a nonthermal distribution similar to that of pickup ions in the solar wind. We predict the H alpha line profiles from the combination of pickup protons and the ordinary shocked protons, and we consider the extent to which this distribution could affect the shock parameters derived from H alpha profiles. The Maxwellian assumption could lead to an underestimate of shock speed by up to about 15%. The isotropization of the pickup ion population generates wave energy, and we find that for the most favorable parameters this energy could significantly heat the thermal particles. Sufficiently accurate profiles could constrain the strength and direction of the magnetic field in the shocked plasma, and we discuss the distortions from a Gaussian profile to be expected in Tycho's supernova remnant.Comment: 13 pages, 6 figure

Spatial distribution of low-energy plasma around 2 comet 67P/CG from Rosetta measurements

International audienceWe use measurements from the Rosetta plasma consortium (RPC) Langmuir probe (LAP) and mutual impedance probe (MIP) to study the spatial distribution of low-energy plasma in the near-nucleus coma of comet 67P/Churyumov-Gerasimenko. The spatial distribution is highly structured with the highest density in the summer hemisphere and above the region connecting the two main lobes of the comet, i.e. the neck region. There is a clear correlation with the neutral density and the plasma to neutral density ratio is found to be ∼1-2·10 −6 , at a cometocentric distance of 10 km and at 3.1 AU from the sun. A clear 6.2 h modulation of the plasma is seen as the neck is exposed twice per rotation. The electron density of the collisonless plasma within 260 km from the nucleus falls of with radial distance as ∼1/r. The spatial structure indicates that local ionization of neutral gas is the dominant source of low-energy plasma around the comet

The defect variance of random spherical harmonics

The defect of a function $f:M\rightarrow \mathbb{R}$ is defined as the difference between the measure of the positive and negative regions. In this paper, we begin the analysis of the distribution of defect of random Gaussian spherical harmonics. By an easy argument, the defect is non-trivial only for even degree and the expected value always vanishes. Our principal result is obtaining the asymptotic shape of the defect variance, in the high frequency limit. As other geometric functionals of random eigenfunctions, the defect may be used as a tool to probe the statistical properties of spherical random fields, a topic of great interest for modern Cosmological data analysis.Comment: 19 page

Some comments on developments in exact solutions in statistical mechanics since 1944

Lars Onsager and Bruria Kaufman calculated the partition function of the Ising model exactly in 1944 and 1949. Since then there have been many developments in the exact solution of similar, but usually more complicated, models. Here I shall mention a few, and show how some of the latest work seems to be returning once again to the properties observed by Onsager and Kaufman.Comment: 28 pages, 5 figures, section on six-vertex model revise

Growing random sequences

We consider the random sequence x[n] = x[n-1] + yxq, with y > 0, where q = 0, 1,..., n - 1 is chosen randomly from a probability distribution P[n] (q). When all q are chosen with equal probability, i.e. P[n](q) = 1/n, we obtain an exact solution for the mean and the divergence of the second moment as functions of n and y. For y = 1 we examine the divergence of the mean value of x[n], as a function of n, for the random sequences generated by power law and exponential P[n](q) and for the non-random sequence P[n](q) = δ[q,β(n-1)]

Holography, Pade Approximants and Deconstruction

We investigate the relation between holographic calculations in 5D and the Migdal approach to correlation functions in large N theories. The latter employs Pade approximation to extrapolate short distance correlation functions to large distances. We make the Migdal/5D relation more precise by quantifying the correspondence between Pade approximation and the background and boundary conditions in 5D. We also establish a connection between the Migdal approach and the models of deconstructed dimensions.Comment: 28 page