Feller Processes: The Next Generation in Modeling. Brownian Motion, L\'evy Processes and Beyond

We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent mixing of L\'evy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also L\'evy processes, of which Brownian Motion is a special case, have become increasingly popular. L\'evy processes are spatially homogeneous, but empirical data often suggest the use of spatially inhomogeneous processes. Thus it seems necessary to go to the next level of generalization: Feller processes. These include L\'evy processes and in particular Brownian motion as special cases but allow spatial inhomogeneities. Many properties of Feller processes are known, but proving the very existence is, in general, very technical. Moreover, an applicable framework for the generation of sample paths of a Feller process was missing. We explain, with practitioners in mind, how to overcome both of these obstacles. In particular our simulation technique allows to apply Monte Carlo methods to Feller processes.Comment: 22 pages, including 4 figures and 8 pages of source code for the generation of sample paths of Feller processe

Boundary correlation function of fixed-to-free bcc operators in square-lattice Ising model

We calculate the boundary correlation function of fixed-to-free boundary condition changing operators in the square-lattice Ising model. The correlation function is expressed in four different ways using $2\times2$ block Toeplitz determinants. We show that these can be transformed into a scalar Toeplitz determinant when the size of the matrix is even. To know the asymptotic behavior of the correlation function at large distance we calculate the asymptotic behavior of this scalar Toeplitz determinant using the Szeg\"o's theorem and the Fisher-Hartwig theorem. At the critical temperature we confirm the power-law behavior of the correlation function predicted by conformal field theory

Embedding into bipartite graphs

The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of $(\frac12+\gamma)n$ when we have the additional structural information of the host graph $G$ being balanced bipartite. This complements results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladk\'y and Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with $r$ and $s$ fixed. Moreover, it implies that the set of Hamilton cycles of $G$ is a generating system for its cycle space.Comment: 16 pages, 2 figure

Metastable precursors during the oxidation of the Ru(0001) surface

Using density-functional theory, we predict that the oxidation of the Ru(0001) surface proceeds via the accumulation of sub-surface oxygen in two-dimensional islands between the first and second substrate layer. This leads locally to a decoupling of an O-Ru-O trilayer from the underlying metal. Continued oxidation results in the formation and stacking of more of these trilayers, which unfold into the RuO_2(110) rutile structure once a critical film thickness is exceeded. Along this oxidation pathway, we identify various metastable configurations. These are found to be rather close in energy, indicating a likely lively dynamics between them at elevated temperatures, which will affect the surface chemical and mechanical properties of the material.Comment: 11 pages including 9 figures. Submitted to Phys. Rev. B. Related publications can be found at http://www.fhi-berlin.mpg.de/th/paper.htm

Out of equilibrium correlations in the XY chain

We study the transversal XY spin-spin correlations in the non-equilibrium steady state constructed in \cite{AP03} and prove their spatial exponential decay close to equilibrium

Interpretation of increased energetic particle flux measurements by SEPT aboard the STEREO spacecraft and contamination

Context. Interplanetary (IP) shocks are known to be accelerators of energetic charged particles observed in-situ in the heliosphere. However, the acceleration of near-relativistic electrons by shocks in the interplanetary medium is often questioned. On 9 August 2011 a Corotating Interaction Region (CIR) passed STEREO B (STB) that resulted in a flux increase in the electron and ion channels of the Solar Electron and Proton Telescope (SEPT). Because electron measurements in the few keV to several 100 keV range rely on the so-called magnet foil technique, which is utilized by SEPT, ions can contribute to the electron channels. Aims. We aim to investigate whether the flux increase in the electron channels of SEPT during the CIR event on 9 August 2011 is caused by ion contamination only. Methods. We compute the SEPT response functions for protons and helium utilizing an updated GEANT4 model of SEPT. The CIR energetic particle ion spectra for protons and helium are assumed to follow a Band function in energy per nucleon with a constant helium to proton ratio. Results. Our analysis leads to a helium to proton ratio of 16.9% and a proton flux following a Band function with the parameters $I_0 = 1.24 \cdot 10^4$ / (cm2 s sr MeV/nuc.), $E_c = 79$ keV/nuc. and spectral indices of $\gamma_1 = -0.94$ and $\gamma_2 = -3.80$ which are in good agreement with measurements by the Suprathermal Ion Telescope (SIT) aboard STB. Conclusions. Since our results explain the SEPT measurements, we conclude that no significant amount of electrons were accelerated between $55$ keV and $425$ keV by the CIR

Opacity in compact extragalactic radio sources and its effect on radio-optical reference frame alignment

Accurate alignment of the radio and optical celestial reference frames requires detailed understanding of physical factors that may cause offsets between the positions of the same object measured in different spectral bands. Opacity in compact extragalactic jets (due to synchrotron self-absorption and external free-free absorption) is one of the key physical phenomena producing such an offset, and this effect is well-known in radio astronomy ("core shift"). We have measured the core shifts in a sample of 29 bright compact extragalactic radio sources observed using very long baseline interferometry (VLBI) at 2.3 and 8.6 GHz. We report the results of these measurements and estimate that the average shift between radio and optical positions of distant quasars would be of the order of 0.1-0.2 mas. This shift exceeds positional accuracy of GAIA and SIM. We suggest two possible approaches to carefully investigate and correct for this effect in order to align accurately the radio and optical positions. Both approaches involve determining a Primary Reference Sample of objects to be used for tying the radio and optical reference frames together.Comment: 4 pages, 1 figure; to appear in IAU Symposium 248 Proceedings, "A Giant Step: from Milli- to Micro-arcsecond Astrometry", eds. W.-J. Jin, I. Platais, M. Perryma

Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle

We prove that for any $n\times n$ matrix, $A$, and $z$ with $|z|\geq \|A\|$, we have that \|(z-A)^{-1}\|\leq\cot (\frac{\pi}{4n}) \dist (z, \spec(A))^{-1}. We apply this result to the study of random orthogonal polynomials on the unit circle.Comment: 27 page

Composition, structure and stability of RuO_2(110) as a function of oxygen pressure

Using density-functional theory (DFT) we calculate the Gibbs free energy to determine the lowest-energy structure of a RuO_2(110) surface in thermodynamic equilibrium with an oxygen-rich environment. The traditionally assumed stoichiometric termination is only found to be favorable at low oxygen chemical potentials, i.e. low pressures and/or high temperatures. At realistic O pressure, the surface is predicted to contain additional terminal O atoms. Although this O excess defines a so-called polar surface, we show that the prevalent ionic model, that dismisses such terminations on electrostatic grounds, is of little validity for RuO_2(110). Together with analogous results obtained previously at the (0001) surface of corundum-structured oxides, these findings on (110) rutile indicate that the stability of non-stoichiometric terminations is a more general phenomenon on transition metal oxide surfaces.Comment: 12 pages including 5 figures. Submitted to Phys. Rev. B. Related publications can be found at http://www.fhi-berlin.mpg.de/th/paper.htm

One-dimensional pair cascade emission in gamma-ray binaries

In gamma-ray binaries such as LS 5039 a large number of electron-positron pairs are created by the annihilation of primary very high energy (VHE) gamma-rays with photons from the massive star. The radiation from these particles contributes to the total high energy gamma-ray flux and can initiate a cascade, decreasing the effective gamma-ray opacity in the system. The aim of this paper is to model the cascade emission and investigate if it can account for the VHE gamma-ray flux detected by HESS from LS 5039 at superior conjunction, where the primary gamma-rays are expected to be fully absorbed. A one-dimensional cascade develops along the line-of-sight if the deflections of pairs induced by the surrounding magnetic field can be neglected. A semi-analytical approach can then be adopted, including the effects of the anisotropic seed radiation field from the companion star. Cascade equations are numerically solved, yielding the density of pairs and photons. In LS 5039, the cascade contribution to the total flux is large and anti-correlated with the orbital modulation of the primary VHE gamma-rays. The cascade emission dominates close to superior conjunction but is too strong to be compatible with HESS measurements. Positron annihilation does not produce detectable 511 keV emission. This study provides an upper limit to cascade emission in gamma-ray binaries at orbital phases where absorption is strong. The pairs are likely to be deflected or isotropized by the ambient magnetic field, which will reduce the resulting emission seen by the observer. Cascade emission remains a viable explanation for the detected gamma-rays at superior conjunction in LS 5039.Comment: 8 pages, 7 figures, 1 table, accepted for publication in Astronomy and Astrophysic