Terrestrial planets across space and time

The study of cosmology, galaxy formation and exoplanets has now advanced to a stage where a cosmic inventory of terrestrial planets may be attempted. By coupling semi-analytic models of galaxy formation to a recipe that relates the occurrence of planets to the mass and metallicity of their host stars, we trace the population of terrestrial planets around both solar-mass (FGK type) and lower-mass (M dwarf) stars throughout all of cosmic history. We find that the mean age of terrestrial planets in the local Universe is $7\pm{}1$ Gyr for FGK hosts and $8\pm{}1$ Gyr for M dwarfs. We estimate that hot Jupiters have depleted the population of terrestrial planets around FGK stars by no more than $\approx 10\%$, and that only $\approx 10\%$ of the terrestrial planets at the current epoch are orbiting stars in a metallicity range for which such planets have yet to be confirmed. The typical terrestrial planet in the local Universe is located in a spheroid-dominated galaxy with a total stellar mass comparable to that of the Milky Way. When looking at the inventory of planets throughout the whole observable Universe, we argue for a total of $\approx 1\times 10^{19}$ and $\approx 5\times 10^{20}$ terrestrial planets around FGK and M stars, respectively. Due to light travel time effects, the terrestrial planets on our past light cone exhibit a mean age of just $1.7\pm 0.2$ Gyr. These results are discussed in the context of cosmic habitability, the Copernican principle and searches for extraterrestrial intelligence at cosmological distances.Comment: 11 pages, 8 figures. v.2: Accepted for publication in ApJ. Some changes in quantitative results compared to v.1, mainly due to differences in IMF assumption

Quicksort with unreliable comparisons: a probabilistic analysis

We provide a probabilistic analysis of the output of Quicksort when comparisons can err.Comment: 29 pages, 3 figure

Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution

We introduce an alternative definition of the relative height h^\kappa(x) of a one-dimensional fluctuating interface indexed by a continuously varying real paramater 0 \leq \kappa \leq 1. It interpolates between the height relative to the initial value (i.e. in x=0) when \kappa = 0 and the height relative to the spatially averaged height for \kappa = 1. We compute exactly the distribution P^\kappa(h_m,L) of the maximum h_m of these relative heights for systems of finite size L and periodic boundary conditions. One finds that it takes the scaling form P^\kappa(h_m,L) = L^{-1/2} f^\kappa (h_m L^{-1/2}) where the scaling function f^\kappa(x) interpolates between the Rayleigh distribution for \kappa=0 and the Airy distribution for \kappa=1, the latter being the probability distribution of the area under a Brownian excursion over the unit interval. For arbitrary \kappa, one finds that it is related to, albeit different from, the distribution of the area restricted to the interval [0, \kappa] under a Brownian excursion over the unit interval.Comment: 25 pages, 4 figure

An exactly solvable self-convolutive recurrence

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform we convert this expression into an explicit, non-recursive solution in which the $n$th coefficient is expressed as the $(n-1)$th moment of a measure, and also as the trace of the $(n-1)$th iterate of a linear operator. Applications of these sequences, and hence of the explicit solution provided, are found in quantum field theory as the number of Feynman diagrams of a certain type and order, in Brownian motion theory, and in combinatorics

A Survey for Massive Giant Planets in Debris Disks with Evacuated Inner Cavities

The commonality of collisionally replenished debris around main sequence stars suggests that minor bodies are frequent around Sun-like stars. Whether or not debris disks in general are accompanied by planets is yet unknown, but debris disks with large inner cavities - perhaps dynamically cleared - are considered to be prime candidates for hosting large-separation massive giant planets. We present here a high-contrast VLT/NACO angular differential imaging survey for eight such cold debris disks. We investigated the presence of massive giant planets in the range of orbital radii where the inner edge of the dust debris is expected. Our observations are sensitive to planets and brown dwarfs with masses >3 to 7 Jupiter mass, depending on the age and distance of the target star. Our observations did not identify any planet candidates. We compare the derived planet mass upper limits to the minimum planet mass required to dynamically clear the inner disks. While we cannot exclude that single giant planets are responsible for clearing out the inner debris disks, our observations constrain the parameter space available for such planets. The non-detection of massive planets in these evacuated debris disks further reinforces the notion that the giant planet population is confined to the inner disk (<15 AU).Comment: Accepted for publication in Ap

Percolation with Multiple Giant Clusters

We study the evolution of percolation with freezing. Specifically, we consider cluster formation via two competing processes: irreversible aggregation and freezing. We find that when the freezing rate exceeds a certain threshold, the percolation transition is suppressed. Below this threshold, the system undergoes a series of percolation transitions with multiple giant clusters ("gels") formed. Giant clusters are not self-averaging as their total number and their sizes fluctuate from realization to realization. The size distribution F_k, of frozen clusters of size k, has a universal tail, F_k ~ k^{-3}. We propose freezing as a practical mechanism for controlling the gel size.Comment: 4 pages, 3 figure

Leadership Statistics in Random Structures

The largest component (``the leader'') in evolving random structures often exhibits universal statistical properties. This phenomenon is demonstrated analytically for two ubiquitous structures: random trees and random graphs. In both cases, lead changes are rare as the average number of lead changes increases quadratically with logarithm of the system size. As a function of time, the number of lead changes is self-similar. Additionally, the probability that no lead change ever occurs decays exponentially with the average number of lead changes.Comment: 5 pages, 3 figure

Reconstruction of Random Colourings

Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random $k$-colourings on the $\Delta$-ary tree for large $k$. Bhatnagar et. al. showed non-reconstruction when $\Delta \leq \frac12 k\log k - o(k\log k)$ and reconstruction when $\Delta \geq k\log k + o(k\log k)$. We tighten this result and show non-reconstruction when $\Delta \leq k[\log k + \log \log k + 1 - \ln 2 -o(1)]$ and reconstruction when $\Delta \geq k[\log k + \log \log k + 1+o(1)]$.Comment: Added references, updated notatio

Consequences of critical interchain couplings and anisotropy on a Haldane chain

Effects of interchain couplings and anisotropy on a Haldane chain have been investigated by single crystal inelastic neutron scattering and density functional theory (DFT) calculations on the model compound SrNi$_2$V$_2$O$_8$. Significant effects on low energy excitation spectra are found where the Haldane gap ($\Delta_0 \approx 0.41J$; where $J$ is the intrachain exchange interaction) is replaced by three energy minima at different antiferromagnetic zone centers due to the complex interchain couplings. Further, the triplet states are split into two branches by single-ion anisotropy. Quantitative information on the intrachain and interchain interactions as well as on the single-ion anisotropy are obtained from the analyses of the neutron scattering spectra by the random phase approximation (RPA) method. The presence of multiple competing interchain interactions is found from the analysis of the experimental spectra and is also confirmed by the DFT calculations. The interchain interactions are two orders of magnitude weaker than the nearest-neighbour intrachain interaction $J$ = 8.7~meV. The DFT calculations reveal that the dominant intrachain nearest-neighbor interaction occurs via nontrivial extended superexchange pathways Ni--O--V--O--Ni involving the empty $d$ orbital of V ions. The present single crystal study also allows us to correctly position SrNi$_2$V$_2$O$_8$ in the theoretical $D$-$J_{\perp}$ phase diagram [T. Sakai and M. Takahashi, Phys. Rev. B 42, 4537 (1990)] showing where it lies within the spin-liquid phase.Comment: 12 pages, 12 figures, 3 tables PRB (accepted). in Phys. Rev. B (2015