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

Parametrically excited "Scars" in Bose-Einstein condensates

Parametric excitation of a Bose-Einstein condensate (BEC) can be realized by periodically changing the interaction strength between the atoms. Above some threshold strength, this excitation modulates the condensate density. We show that when the condensate is trapped in a potential well of irregular shape, density waves can be strongly concentrated ("scarred") along the shortest periodic orbits of a classical particle moving within the confining potential. While single-particle wave functions of systems whose classical counterpart is chaotic may exhibit rich scarring patterns, in BEC, we show that nonlinear effects select mainly those scars that are locally described by stripes. Typically, these are the scars associated with self retracing periodic orbits that do not cross themselves in real space. Dephasing enhances this behavior by reducing the nonlocal effect of interference

Gravitational wave energy spectrum of a parabolic encounter

We derive an analytic expression for the energy spectrum of gravitational waves from a parabolic Keplerian binary by taking the limit of the Peters and Matthews spectrum for eccentric orbits. This demonstrates that the location of the peak of the energy spectrum depends primarily on the orbital periapse rather than the eccentricity. We compare this weak-field result to strong-field calculations and find it is reasonably accurate (~10%) provided that the azimuthal and radial orbital frequencies do not differ by more than ~10%. For equatorial orbits in the Kerr spacetime, this corresponds to periapse radii of rp > 20M. These results can be used to model radiation bursts from compact objects on highly eccentric orbits about massive black holes in the local Universe, which could be detected by LISA.Comment: 5 pages, 3 figures. Minor changes to match published version; figure 1 corrected; references adde

New Measurements of Venus Winds with Ground-Based Doppler Velocimetry at CFHT

operations with observations from the ground using various techniques and spectral domains (Lellouch and Witasse, 2008). We present an analysis of Venus Doppler winds at cloud tops based on observations made at the Canada France Hawaii 3.6-m telescope (CFHT) with the ESPaDOnS visible spectrograph. These observations consisted of high-resolution spectra of Fraunhofer lines in the visible range (0.37-1.05 μm) to measure the winds at cloud tops using the Doppler shift of solar radiation scattered by cloud top particles in the observer's direction (Widemann et al., 2007, 2008). The observations were made during 19-20 February 2011 and were coordinated with Visual Monitoring Camera (VMC) observations by Venus Express. The complete optical spectrum was collected over 40 spectral orders at each point with 2-5 seconds exposures, at a resolution of about 80000. The observations included various points of the dayside hemisphere at a phase angle of 67°, between +10° and -60° latitude, in steps of 10° , and from +70° to -12° longitude relative to sub-Earth meridian in steps of 12°. The Doppler shift measured in scattered solar light on Venus dayside results from two instantaneous motions: (1) a motion between the Sun and Venus upper cloud particles; (2) a motion between the observer and Venus clouds. The measured Doppler shift, which results from these two terms combined, varies with the planetocentric longitude and latitude and is minimum at meridian ΦN = ΦSun - ΦEarth where the two components subtract to each other for a pure zonal regime. Due to the need for maintaining a stable velocity reference during the course of acquisition using high resolution spectroscopy, we measure relative Doppler shifts to ΦN. The main purpose of our work is to provide variable wind measurements with respect to the background atmosphere, complementary to simultaneous measurements made with the VMC camera onboard the Venus Express. We will present first results from this work, comparing with previous results by the CFHT/ESPaDOnS and VLT-UVES spectrographs (Machado et al., 2012), with Galileo fly-by measurements and with VEx nominal mission observations (Peralta et al., 2007, Luz et al., 2011). Acknowledgements: The authors acknowledge support from FCT through projects PTDC/CTE-AST/110702/2009 and PEst-OE/FIS/UI2751/2011. PM and TW also acknowledge support from the Observatoire de Paris. Lellouch, E., and Witasse, O., A coordinated campaign of Venus ground-based observations and Venus Express measurements, Planetary and Space Science 56 (2008) 1317-1319. Luz, D., et al., Venus's polar vortex reveals precessing circulation, Science 332 (2011) 577-580. Machado, P., Luz, D. Widemann, T., Lellouch, E., Witasse, O, Characterizing the atmospheric dynamics of Venus from ground-based Doppler velocimetry, Icarus, submitted. Peralta J., R. Hueso, A. Sánchez-Lavega, A reanalysis of Venus winds at two cloud levels from Galileo SSI images, Icarus 190 (2007) 469-477. Widemann, T., Lellouch, E., Donati, J.-F., 2008, Venus Doppler winds at Cloud Tops Observed with ESPaDOnS at CFHT, Planetary and Space Science, 56, 1320-1334

Geometric gauge potentials and forces in low-dimensional scattering systems

We introduce and analyze several low-dimensional scattering systems that exhibit geometric phase phenomena. The systems are fully solvable and we compare exact solutions of them with those obtained in a Born-Oppenheimer projection approximation. We illustrate how geometric magnetism manifests in them, and explore the relationship between solutions obtained in the diabatic and adiabatic pictures. We provide an example, involving a neutral atom dressed by an external field, in which the system mimics the behavior of a charged particle that interacts with, and is scattered by, a ferromagnetic material. We also introduce a similar system that exhibits Aharonov-Bohm scattering. We propose some practical applications. We provide a theoretical approach that underscores universality in the appearance of geometric gauge forces. We do not insist on degeneracies in the adiabatic Hamiltonian, and we posit that the emergence of geometric gauge forces is a consequence of symmetry breaking in the latter.Comment: (Final version, published in Phy. Rev. A. 86, 042704 (2012

Quantum Spin Hall Effect and Topologically Invariant Chern Numbers

We present a topological description of quantum spin Hall effect (QSHE) in a two-dimensional electron system on honeycomb lattice with both intrinsic and Rashba spin-orbit couplings. We show that the topology of the band insulator can be characterized by a $2\times 2$ traceless matrix of first Chern integers. The nontrivial QSHE phase is identified by the nonzero diagonal matrix elements of the Chern number matrix (CNM). A spin Chern number is derived from the CNM, which is conserved in the presence of finite disorder scattering and spin nonconserving Rashba coupling. By using the Laughlin's gedanken experiment, we numerically calculate the spin polarization and spin transfer rate of the conducting edge states, and determine a phase diagram for the QSHE.Comment: 4 pages and 4 figure

Nonclassical Degrees of Freedom in the Riemann Hamiltonian

The Hilbert-Polya conjecture states that the imaginary parts of the zeros of the Riemann zeta function are eigenvalues of a quantum hamiltonian. If so, conjectures by Katz and Sarnak put this hamiltonian in Altland and Zirnbauer's universality class C. This implies that the system must have a nonclassical two-valued degree of freedom. In such a system, the dominant primitive periodic orbits contribute to the density of states with a phase factor of -1. This resolves a previously mysterious sign problem with the oscillatory contributions to the density of the Riemann zeros.Comment: 4 pages, no figures; v3-6 have minor corrections to v2, v2 has a more complete solution of the sign problem than v

Can apparent superluminal neutrino speeds be explained as a quantum weak measurement?

Probably not.Comment: 10 pages, 1 figur

Classical and quantum chaos in a circular billiard with a straight cut

We study classical and quantum dynamics of a particle in a circular billiard with a straight cut. This system can be integrable, nonintegrable with soft chaos, or nonintegrable with hard chaos, as we vary the size of the cut. We use a quantum web to show differences in the quantum manifestations of classical chaos for these three different regimes.Comment: LaTeX2e, 8 pages including 3 Postscript figures and 4 GIF figures, submitted to Phys. Rev.