Formulas for ASEP with Two-Sided Bernoulli Initial Condition

For the asymmetric simple exclusion process on the integer lattice with two-sided Bernoulli initial condition, we derive exact formulas for the following quantities: (1) the probability that site x is occupied at time t; (2) a correlation function, the probability that site 0 is occupied at time 0 and site x is occupied at time t; (3) the distribution function for the total flux across 0 at time t and its exponential generating function.Comment: 18 page

From Random Matrices to Stochastic Operators

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.Comment: 41 pages, 5 figures. Submitted to Journal of Statistical Physics. Changes in this revision: recomputed Monte Carlo simulations, added reference [19], fit into margins, performed minor editin

CMB Constraints on WIMP Annihilation: Energy Absorption During the Recombination Epoch

We compute in detail the rate at which energy injected by dark matter annihilation heats and ionizes the photon-baryon plasma at z ~ 1000, and provide accurate fitting functions over the relevant redshift range for a broad array of annihilation channels and DM masses. The resulting perturbations to the ionization history can be constrained by measurements of the CMB temperature and polarization angular power spectra. We show that models which fit recently measured excesses in 10-1000 GeV electron and positron cosmic rays are already close to the 95% confidence limits from WMAP. The recently launched Planck satellite will be capable of ruling out a wide range of DM explanations for these excesses. In models of dark matter with Sommerfeld-enhanced annihilation, where sigma v rises with decreasing WIMP velocity until some saturation point, the WMAP5 constraints imply that the enhancement must be close to saturation in the neighborhood of the Earth.Comment: 17 pages, 6 figures, v2 extends discussion of constraints on Sommerfeld-enhanced model

Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles

The probabilities for gaps in the eigenvalue spectrum of the finite dimension $N \times N$ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second and third order nonlinear ordinary differential equations defining the probabilities in the general $N$ case. For N=1 and N=2 the probabilities and thus the solution of the equations are given explicitly. An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as $N \to \infty$, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a {Painlev\'e-V} transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a {Painlev\'e-VI} transcendent.Comment: 32 pages, Latex2

A Fredholm Determinant Representation in ASEP

In previous work the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice. The dynamics are uniquely determined once the initial state is specified. In this note we restrict our attention to the case of step initial condition with particles at the positive integers, and consider the distribution function for the m'th particle from the left. In the previous work an infinite series of multiple integrals was derived for this distribution. In this note we show that the series can be summed to give a single integral whose integrand involves a Fredholm determinant. We use this determinant representation to derive (non-rigorously, at this writing) a scaling limit.Comment: 12 Pages. Version 3 includes a scaling conjectur

Finite time corrections in KPZ growth models

We consider some models in the Kardar-Parisi-Zhang universality class, namely the polynuclear growth model and the totally/partially asymmetric simple exclusion process. For these models, in the limit of large time t, universality of fluctuations has been previously obtained. In this paper we consider the convergence to the limiting distributions and determine the (non-universal) first order corrections, which turn out to be a non-random shift of order t^{-1/3} (of order 1 in microscopic units). Subtracting this deterministic correction, the convergence is then of order t^{-2/3}. We also determine the strength of asymmetry in the exclusion process for which the shift is zero. Finally, we discuss to what extend the discreteness of the model has an effect on the fitting functions.Comment: 34 pages, 5 figures, LaTeX; Improved version including shift of PASEP height functio

On the Distribution of a Second Class Particle in the Asymmetric Simple Exclusion Process

We give an exact expression for the distribution of the position X(t) of a single second class particle in the asymmetric simple exclusion process (ASEP) where initially the second class particle is located at the origin and the first class particles occupy the sites {1,2,...}

Asymptotic Level Spacing of the Laguerre Ensemble: A Coulomb Fluid Approach

We determine the asymptotic level spacing distribution for the Laguerre Ensemble in a single scaled interval, $(0,s)$, containing no levels, E_{\bt}(0,s), via Dyson's Coulomb Fluid approach. For the $\alpha=0$ Unitary-Laguerre Ensemble, we recover the exact spacing distribution found by both Edelman and Forrester, while for $\alpha\neq 0$, the leading terms of $E_{2}(0,s)$, found by Tracy and Widom, are reproduced without the use of the Bessel kernel and the associated Painlev\'e transcendent. In the same approximation, the next leading term, due to a ``finite temperature'' perturbation (\bt\neq 2), is found.Comment: 10pp, LaTe

The Dynamical State of Barnard 68: A Thermally Supported, Pulsating Dark Cloud

We report sensitive, high resolution molecular-line observations of the dark cloud Barnard 68 obtained with the IRAM 30-m telescope. We analyze spectral-line observations of C18O, CS(2--1), C34S(2--1), and N2H+(1--0) in order to investigate the kinematics and dynamical state of the cloud. We find extremely narrow linewidths in the central regions of the cloud. These narrow lines are consistent with thermally broadened profiles for the measured gas temperature of 10.5 K. We determine the thermal pressure to be a factor 4 -- 5 times greater than the non-thermal (turbulent) pressure in the central regions of the cloud, indicating that thermal pressure is the primary source of support against gravity in this cloud. This confirms the inference of a thermally supported cloud drawn previously from deep infrared extinction measurements. The rotational kinetic energy is found to be only a few percent of the gravitational potential energy, indicating that the contribution of rotation to the overall stability of the cloud is insignificant. Finally, our observations show that CS line is optically thick and self-reversed across nearly the entire projected surface of the cloud. The shapes of the self-reversed profiles are asymmetric and are found to vary across the cloud in such a manner that the presence of both inward and outward motions are observed within the cloud. Moreover, these motions appear to be globally organized in a clear and systematic alternating spatial pattern which is suggestive of a small amplitude, non-radial oscillation or pulsation of the outer layers of the cloud about an equilibrium configuration.Comment: To appear in the Astrophysical Journal; 23 pages, 8 figures; Manuscript and higher resolution images can be obtained at http://cfa-www.harvard.edu/~ebergin/pubs_html/b68_vel.htm