Martin boundary of a reflected random walk on a half-space

The complete representation of the Martin compactification for reflected random walks on a half-space $\Z^d\times\N$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the ``radial'' compactification obtained by Ney and Spitzer for the homogeneous random walks in $\Z^d$ : convergence of a sequence of points $z_n\in\Z^{d-1}\times\N$ to a point of on the Martin boundary does not imply convergence of the sequence $z_n/|z_n|$ on the unit sphere $S^d$. Our approach relies on the large deviation properties of the scaled processes and uses Pascal's method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.Comment: 42 pages, preprint, CNRS UMR 808

Survival, extinction and approximation of discrete-time branching random walks

We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.Comment: 32 pages, a few misprints have been correcte

Orbital parameters, masses and distance to Beta Centauri determined with the Sydney University Stellar Interferometer and high resolution spectroscopy

The bright southern binary star beta Centauri (HR 5267) has been observed with the Sydney University Stellar Interferometer (SUSI) and spectroscopically with the ESO CAT and Swiss Euler telescopes at La Silla. The interferometric observations have confirmed the binary nature of the primary component and have enabled the determination of the orbital parameters of the system. At the observing wavelength of 442 nm the two components of the binary system have a magnitude difference of 0.15. The combination of interferometric and spectroscopic data gives the following results: orbital period 357 days, semi-major axis 25.30 mas, inclination 67.4 degrees, eccentricity 0.821, distance 102.3 pc, primary and secondary masses M1 = M2 = 9.1 solar masses and absolute visual magnitudes of the primary and secondary M1V = -3.85 and M2V = -3.70. The high accuracy of the results offers a fruitful starting point for future asteroseismic modelling of the pulsating binary components.Comment: 10 pages, 4 figures. Accepted for publication in MNRA

Boundaries of Disk-like Self-affine Tiles

Let $T:= T(A, {\mathcal D})$ be a disk-like self-affine tile generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\mathcal D}$, and let $f(x)=x^{2}+px+q$ be the characteristic polynomial of $A$. In the paper, we identify the boundary $\partial T$ with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,{\mathcal D})$ to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\omega$, we find the generalized Hausdorff dimension $\dim_H^{\omega} (\partial T)=2\log \rho(M)/\log |q|$ where $\rho(M)$ is the spectral radius of certain contact matrix $M$. Especially, when $A$ is a similarity, we obtain the standard Hausdorff dimension $\dim_H (\partial T)=2\log \rho/\log |q|$ where $\rho$ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$, which is simpler than the known result.Comment: 26 pages, 11 figure

Orbital elements, masses and distance of lambda Scorpii A and B determined with the Sydney University Stellar Interferometer and high resolution spectroscopy

The triple system HD158926 (lambda Sco) has been observed interferometrically with the Sydney University Stellar Interferometer and the elements of the wide orbit have been determined. These are significantly more accurate than the previous elements found spectroscopically. The inclination of the wide orbit is consistent with the inclination previously found for the orbit of the close companion. The wide orbit also has low eccentricity, suggesting that the three stars were formed at the same time. The brightness ratio between the two B stars was also measured at lambda = 442nm and 700nm. The brightness ratio and colour index are consistent with the previous classification of lambda Sco A as B1.5 and lambda Sco B as B2. Evolutionary models show that the two stars lie on the main sequence. Since they have have the same age and luminosity class (IV) the mass-luminosity relation can be used to determine the mass ratio of the two stars: M_B/M_A = 0.76+/-0.04. The spectroscopic data have been reanalyzed using the interferometric values for P, T, e and omega, leading to revised values for a_1sin i and the mass function. The individual masses can be found from the mass ratio, the mass function, spectrum synthesis and the requirement that the age of both components must be the same: M_A = 10.4+/-1.3 Msun and M_B = 8.1+/-1.0 Msun. The masses, angular semimajor axis and the period of the system can be used to determine the dynamical parallax. We find the distance to lambda Sco to be 112+/-5 pc, which is approximately a factor of two closer than the HIPPARCOS value of 216+/-42 pc.Comment: 8 pages, 4 figures. Accepted for publication by Monthly Notices of the Royal Astronomical Societ

Random tree growth by vertex splitting

We study a model of growing planar tree graphs where in each time step we separate the tree into two components by splitting a vertex and then connect the two pieces by inserting a new link between the daughter vertices. This model generalises the preferential attachment model and Ford's $\alpha$-model for phylogenetic trees. We develop a mean field theory for the vertex degree distribution, prove that the mean field theory is exact in some special cases and check that it agrees with numerical simulations in general. We calculate various correlation functions and show that the intrinsic Hausdorff dimension can vary from one to infinity, depending on the parameters of the model.Comment: 47 page

Nonequilibrium stationary states and equilibrium models with long range interactions

It was recently suggested by Blythe and Evans that a properly defined steady state normalisation factor can be seen as a partition function of a fictitious statistical ensemble in which the transition rates of the stochastic process play the role of fugacities. In analogy with the Lee-Yang description of phase transition of equilibrium systems, they studied the zeroes in the complex plane of the normalisation factor in order to find phase transitions in nonequilibrium steady states. We show that like for equilibrium systems, the ``densities'' associated to the rates are non-decreasing functions of the rates and therefore one can obtain the location and nature of phase transitions directly from the analytical properties of the ``densities''. We illustrate this phenomenon for the asymmetric exclusion process. We actually show that its normalisation factor coincides with an equilibrium partition function of a walk model in which the ``densities'' have a simple physical interpretation.Comment: LaTeX, 23 pages, 3 EPS figure

Multi-agent Coordination in Directed Moving Neighborhood Random Networks

In this paper, we consider the consensus problem of dynamical multiple agents that communicate via a directed moving neighborhood random network. Each agent performs random walk on a weighted directed network. Agents interact with each other through random unidirectional information flow when they coincide in the underlying network at a given instant. For such a framework, we present sufficient conditions for almost sure asymptotic consensus. Some existed consensus schemes are shown to be reduced versions of the current model.Comment: 9 page

Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensemble

The paper studies the spectral properties of large Wigner, band and sample covariance random matrices with heavy tails of the marginal distributions of matrix entries.Comment: This is an extended version of my talk at the QMath 9 conference at Giens, France on September 13-17, 200

On the order of summability of the Fourier inversion formula

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems