One-dimensional Voter Model Interface Revisited

We consider the voter model on Z, starting with all 1's to the left of the origin and all 0's to the right of the origin. It is known that if the associated random walk kernel p has zero mean and a finite r-th moment for any r>3, then the evolution of the boundaries of the interface region between 1's and 0's converge in distribution to a standard Brownian motion (B_t)_{t>0} under diffusive scaling of space and time. This convergence fails when p has an infinite r-th moment for any r<3, due to the loss of tightness caused by a few isolated 1's appearing deep within the regions of all 0's (and vice versa) at exceptional times. In this note, we show that as long as p has a finite second moment, the measure-valued process induced by the rescaled voter model configuration is tight, and converges weakly to the measure-valued process 1_{x0.Comment: revised versio

Critical age-dependent branching Markov processes and their scaling limits

This paper studies: (i) the long-time behaviour of the empirical distribution of age and normalized position of an age-dependent critical branching Markov process conditioned on non-extinction; and (ii) the super-process limit of a sequence of age dependent critical branching Brownian motions

Infinite dimensional stochastic differential equations of Ornstein-Uhlenbeck type

We consider the operator \sL f(x)=\tfrac12 \sum_{i,j=1}^\infty a_{ij}(x)\frac{\del^2 f}{\del x_i \del x_j}(x)-\sum_{i=1}^\infty \lam_i x_i b_i(x) \frac{\del f}{\del x_i}(x). We prove existence and uniqueness of solutions to the martingale problem for this operator under appropriate conditions on the $a_{ij}, b_i$, and \lam_i. The process corresponding to \sL solves an infinite dimensional stochastic differential equation similar to that for the infinite dimensional Ornstein-Uhlenbeck process

Atomic and Molecular Absorption in Redshifted Radio Sources

We report on a survey for associated HI 21-cm and OH 18-cm absorption with the Giant Metrewave Radio Telescope at redshifts z = 0.2-0.4. Although the low redshift selection ensures that our targets are below the critical ultra-violet luminosity, which is hypothesised to ionise all of the neutral gas in the host galaxy, we do not obtain any detections in the six sources searched. Analysing these in context of the previous surveys, in addition to the anti-correlation with the ultra-violet luminosity (ionising photon rate), we find a correlation between the strength of the absorption and the blue -- near-infrared colour, as well as the radio-band turnover frequency. We believe that these are due to the photo-ionisation of the neutral gas, an obscured sight-line being more conducive to the presence of cold gas and the compact radio emission being better intercepted by the absorbing gas, maximising the flux coverage, respectively. Regarding the photo-ionisation, the compilation of the previous surveys increases the significance of the critical ionising photon rate, above which all of the gas in the host galaxy is hypothesised to be ionised, to >5 sigma. This reaffirms that this is an ubiquitous effect, which has profound implications for the detection of neutral gas in these objects with the Square Kilometre Array.Comment: Accepted by MNRA

The Extended GMRT Radio Halo Survey I: New upper limits on radio halos and mini-halos

A fraction of galaxy clusters host diffuse radio sources called radio halos, radio relics and mini-halos. We present the sample and first results from the Extended GMRT Radio Halo Survey (EGRHS)- an extension of the GMRT Radio Halo Survey (GRHS, Venturi et al. 2007, 2008). It is a systematic radio survey of galaxy clusters selected from the REFLEX and eBCS X-ray catalogs . Analysis of GMRT data at 610/ 235/ 325 MHz on 12 galaxy clusters are presented. We report the detection of a newly discovered mini-halo in the cluster RXJ1532.9+3021 at 610 MHz. A small scale relic (~200 kpc) is suspected in the cluster Z348. We do not detect cluster-scale diffuse emission in 11 clusters. Robust upper limits on the detection of radio halo of size of 1 Mpc are determined. We also present upper limits on the detections of mini-halos in a sub-sample of cool-core clusters. The upper limits for radio halos and mini-halos are plotted in the radio power- X-ray luminosity plane and the correlations are discussed. Diffuse extended emission, not related to the target clusters, but detected as by-products in the sensitive images of two of the cluster fields (A689 and RXJ0439.0+0715) are reported. Based on the information about the presence of radio halos (or upper limits), available on 48 clusters out of the total sample of 67 clusters (EGRHS+GRHS), we find that ~23% of the clusters host radio halos. The radio halo fraction rises to ~31%, when only the clusters with X-ray luminosities >8x10^44 erg/s are considered. Mini-halos are found in ~50 % of cool-core clusters. A qualitative examination of the X-ray images of the clusters with no diffuse radio emission indicates that a majority of these clusters do not show extreme dynamical disturbances and supports the idea that mergers play an important role in the generation of radio halos/relics.Comment: 21 pages, 18 figures, 3 tables, accepted for publication in A&

The Extended GMRT Radio Halo Survey II: Further results and analysis of the full sample

The intra-cluster medium contains cosmic rays and magnetic fields that are manifested through the large scale synchrotron sources, termed as radio halos, relics and mini-halos. The Extended Giant Metrewave Radio Telescope (GMRT) Radio Halo Survey (EGRHS) is an extension of the GMRT Radio Halo Survey (GRHS) designed to search for radio halos using GMRT 610/235 MHz observations. The GRHS+EGRHS consists of 64 clusters in the redshift range 0.2 -- 0.4 that have an X-ray luminosity larger than 5x10^44 erg/s in the 0.1 -- 2.4 keV band and with declinations > -31 deg in the REFLEX and eBCS X-ray cluster catalogues. In this second paper in the series, GMRT 610/235 MHz data on the last batch of 11 galaxy clusters and the statistical analysis of the full sample are presented. A new mini-halo in RXJ2129.6+0005 and candidate diffuse sources in Z5247, A2552 and Z1953 are discovered. A unique feature of this survey are the upper limits on the detections of 1 Mpc sized radio halos; 4 new are presented here making a total of 31 in the survey. Of the sample, 58 clusters that have adequately sensitive radio information were used to obtain the most accurate occurrence fractions so far. The occurrence of radio halos in our X-ray selected sample is ~22%, that of mini-halos is 13% and that of relics is ~5%. The radio power - X-ray luminosity diagrams for the radio halos and mini-halos with the detections and upper limits are presented. The morphological estimators namely, centroid shift (w), concentration parameter (c) and power ratios (P_3/P_0) derived from the Chandra X-ray images are used as proxies for the dynamical states of the GRHS+EGRHS clusters. The clusters with radio halos and mini-halos occupy distinct quadrants in the c-w, c-P_3/P_0 and w - P_3/P_0 planes, corresponding to the more and less morphological disturbance, respectively. The non-detections span both the quadrants.Comment: 24 pages, 5 tables, 25 figures, accepted for publication in A&