Critical random graphs: limiting constructions and distributional properties

We consider the Erdos-Renyi random graph G(n,p) inside the critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous paper (arXiv:0903.4730) that considering the connected components of G(n,p) as a sequence of metric spaces with the graph distance rescaled by n^{-1/3} and letting n go to infinity yields a non-trivial sequence of limit metric spaces C = (C_1, C_2, ...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R_+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak, Pittel and Wierman by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component.Comment: 30 pages, 4 figure

Critical random graphs : limiting constructions and distributional properties

We consider the Erdos-Renyi random graph G(n, p) inside the critical window, where p = 1/n + lambda n(-4/3) for some lambda is an element of R. We proved in Addario-Berry et al. [2009+] that considering the connected components of G(n, p) as a sequence of metric spaces with the graph distance rescaled by n(-1/3) and letting n -> infinity yields a non-trivial sequence of limit metric spaces C = (C-1, C-2,...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak et al. [1994] by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component

Quantum Monte Carlo simulations of a particle in a random potential

In this paper we carry out Quantum Monte Carlo simulations of a quantum particle in a one-dimensional random potential (plus a fixed harmonic potential) at a finite temperature. This is the simplest model of an interface in a disordered medium and may also pertain to an electron in a dirty metal. We compare with previous analytical results, and also derive an expression for the sample to sample fluctuations of the mean square displacement from the origin which is a measure of the glassiness of the system. This quantity as well as the mean square displacement of the particle are measured in the simulation. The similarity to the quantum spin glass in a transverse field is noted. The effect of quantum fluctuations on the glassy behavior is discussed.Comment: 23 pages, 7 figures included as eps files, uses RevTeX. Accepted for publication in J. of Physics A: Mathematical and Genera

Large time dynamics and aging of a polymer chain in a random potential

We study the out-of-equilibrium large time dynamics of a gaussian polymer chain in a quenched random potential. The dynamics studied is a simple Langevin dynamics commonly referred to as the Rouse model. The equations for the two-time correlation and response function are derived within the gaussian variational approximation. In order to implement this approximation faithfully, we employ the supersymmetric representation of the Martin-Siggia-Rose dynamical action. For a short ranged correlated random potential the equations are solved analytically in the limit of large times using certain assumptions concerning the asymptotic behavior. Two possible dynamical behaviors are identified depending upon the time separation- a stationary regime and an aging regime. In the stationary regime time translation invariance holds and so is the fluctuation dissipation theorem. The aging regime which occurs for large time separations of the two-time correlation functions is characterized by history dependence and the breakdown of certain equilibrium relations. The large time limit of the equations yields equations among the order parameters that are similar to the equations obtained in the statics using replicas. In particular the aging solution corresponds to the broken replica solution. But there is a difference in one equation that leads to important consequences for the solution. The stationary regime corresponds to the motion of the polymer inside a local minimum of the random potential, whereas in the aging regime the polymer hops between different minima. As a byproduct we also solve exactly the dynamics of a chain in a random potential with quadratic correlations.Comment: 21 pages, RevTeX

Logarithmic roughening in a growth process with edge evaporation

Roughening transitions are often characterized by unusual scaling properties. As an example we investigate the roughening transition in a solid-on-solid growth process with edge evaporation [Phys. Rev. Lett. 76, 2746 (1996)], where the interface is known to roughen logarithmically with time. Performing high-precision simulations we find appropriate scaling forms for various quantities. Moreover we present a simple approximation explaining why the interface roughens logarithmically.Comment: revtex, 6 pages, 7 eps figure

NICMOS Observations of Low-Redshift Quasar Host Galaxies

We have obtained Near-Infrared Camera and Multi-Object Spectrometer images of 16 radio quiet quasars observed as part of a project to investigate the ``luminosity/host-mass limit.'' The limit results were presented in McLeod, Rieke, & Storrie-Lombardi (1999). In this paper, we present the images themselves, along with 1- and 2-dimensional analyses of the host galaxy properties. We find that our model-independent 1D technique is reliable for use on ground-based data at low redshifts; that many radio-quiet quasars live in deVaucouleurs-law hosts, although some of the techniques used to determine host type are questionable; that complex structure is found in many of the hosts, but that there are some hosts that are very smooth and symmetric; and that the nuclei radiate at ~2-20% of the Eddington rate based on the assumption that all galaxies have central black holes with a constant mass fraction of 0.6%. Despite targeting hard-to-resolve hosts, we have failed to find any that imply super-Eddington accretion rates.Comment: To appear in ApJ, 28 pages including degraded figures. Download the paper with full-resolutio figures from http://www.astro.wellesley.edu/kmcleod/mm.p

Langevin Dynamics of the vortex matter two-stage melting transition in Bi_2Sr_2CaCu_2O8+δ_{8+\delta} in the presence of straight and of tilted columnar defects

In this paper we use London Langevin molecular dynamics simulations to investigate the vortex matter melting transition in the highly anisotropic high-temperature superconductor material Bi_2Sr_2CaCu_2O$_{8+\delta}$ in the presence of low concentration of columnar defects (CDs). We reproduce with further details our previous results obtained by using Multilevel Monte Carlo simulations that showed that the melting of the nanocrystalline vortex matter occurs in two stages: a first stage melting into nanoliquid vortex matter and a second stage delocalization transition into a homogeneous liquid. Furthermore, we report on new dynamical measurements in the presence of a current that identifies clearly the irreversibility line and the second stage delocalization transition. In addition to CDs aligned along the c-axis we also simulate the case of tilted CDs which are aligned at an angle with respect to the applied magnetic field. Results for CDs tilted by $45^{\circ}$ with respect to c-axis show that the locations of the melting and delocalization transitions are not affected by the tilt when the ratio of flux lines to CDs remains constant. On the other hand we argue that some dynamical properties and in particular the position of the irreversibility line should be affected.Comment: 13 pages, 11 figure

A comparison of the optical properties of radio-loud and radio-quiet quasars

We have made radio observations of 87 optically selected quasars at 5 GHz with the VLA in order to measure the radio power for these objects and hence determine how the fraction of radio-loud quasars varies with redshift and optical luminosity. The sample has been selected from the recently completed Edinburgh Quasar Survey and covers a redshift range of 0.3 < z < 1.5 and an optical absolute magnitude range of -26.5 < M_{B} < -23.5 (h, q_{0} = 1/2). We have also matched up other existing surveys with the FIRST and NVSS radio catalogues and combined these data so that the optical luminosity-redshift plane is now far better sampled than previously. We have fitted a model to the probability of a quasar being radio-loud as a function of absolute magnitude and redshift and from this model infer the radio-loud and radio-quiet optical luminosity functions. The radio-loud optical luminosity function is featureless and flatter than the radio-quiet one. It evolves at a marginally slower rate if quasars evolve by density evolution, but the difference in the rate of evolutions of the two different classes is much less than was previously thought. We show, using Monte-Carlo simulations, that the observed difference in the shape of the optical luminosity functions can be partly accounted for by Doppler boosting of the optical continuum of the radio-loud quasars and explain how this can be tested in the future.Comment: 33 pages, 9 postscript figures, uses the AAS aaspp4 LaTeX style file, to appear in the 1 February 1999 issue of The Astrophysical Journa