Noncolliding Squared Bessel Processes

We consider a particle system of the squared Bessel processes with index $\nu > -1$ conditioned never to collide with each other, in which if $-1 < \nu < 0$ the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function $J_{\nu}$ is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.Comment: v3: LaTeX2e, 26 pages, no figure, corrections made for publication in J. Stat. Phy

A real quaternion spherical ensemble of random matrices

One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form \bY= \bA^{-1} \bB, where \bA and \bB are independent $N\times N$ matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensembles, the eigenvalue jpdf and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices $\beta=1,2,4$. We find that the eigenvalue density (after stereographic projection onto the sphere) has a depletion of eigenvalues along a ring corresponding to the real axis, with reflective symmetry about this ring. However, in the limit of large matrix dimension, this eigenvalue density approaches that of the corresponding complex ensemble, a density which is uniform on the sphere. This result is in keeping with the spherical law (analogous to the circular law for iid matrices), which states that for matrices having the spherical structure \bY= \bA^{-1} \bB, where \bA and \bB are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.Comment: 25 pages, 3 figures. Added another citation in version

Moderate deviations via cumulants

The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples of random objects we treat include dependency graphs, subgraph-counting statistics in Erd\H{o}s-R\'enyi random graphs and $U$-statistics. Moreover, we prove moderate deviation principles for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices as well as the number of particles in a growing box of random determinantal point processes like the number of eigenvalues in the GUE or the number of points in Airy, Bessel, and $\sin$ random point fields.Comment: 24 page

Long-term perturbations due to a disturbing body in elliptic inclined orbit

In the current study, a double-averaged analytical model including the action of the perturbing body's inclination is developed to study third-body perturbations. The disturbing function is expanded in the form of Legendre polynomials truncated up to the second-order term, and then is averaged over the periods of the spacecraft and the perturbing body. The efficiency of the double-averaged algorithm is verified with the full elliptic restricted three-body model. Comparisons with the previous study for a lunar satellite perturbed by Earth are presented to measure the effect of the perturbing body's inclination, and illustrate that the lunar obliquity with the value 6.68\degree is important for the mean motion of a lunar satellite. The application to the Mars-Sun system is shown to prove the validity of the double-averaged model. It can be seen that the algorithm is effective to predict the long-term behavior of a high-altitude Martian spacecraft perturbed by Sun. The double-averaged model presented in this paper is also applicable to other celestial systems.Comment: 28 pages, 6 figure

Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an $n \times n$ matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let $x_k$ denote eigenvalue number $k$. Under the condition that both $k$ and $n-k$ tend to infinity with $n$, we show that $x_k$ is normally distributed in the limit. We also consider the joint limit distribution of $m$ eigenvalues from the GOE or GSE with similar conditions on the indices. The result is an $m$-dimensional normal distribution. Using a recent universality result by Tao and Vu, we extend our results to a class of Wigner real symmetric matrices with non-Gaussian entries that have an exponentially decaying distribution and whose first four moments match the Gaussian moments.Comment: 21 pages, to appear, J. Stat. Phys. References and other corrections suggested by the referees have been incorporate

Cell shape analysis of random tessellations based on Minkowski tensors

To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic models. In the context of image analysis of synthetic and biological materials, this question is central to the problem of inferring information about formation processes from spatial measurements of resulting random structures. We address this question by a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models. We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data are complemented by mechanically stable crystalline sphere and disordered ellipsoid packings and area-minimising foam models. We find that shape indices of individual cells are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of the shape indices between many of these tessellation models. Given a realization of a tessellation, these shape indices can narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by density-resolved volume-shape correlations.Comment: Chapter of the forthcoming book "Tensor Valuations and their Applications in Stochastic Geometry and Imaging" in Lecture Notes in Mathematics edited by Markus Kiderlen and Eva B. Vedel Jense

The Alvarez impact theory of mass extinction; limits to its applicability and the „great expectations syndrome”

For the past three decades, the Alvarez impact theory of mass extinction, causally related to catastrophic meteorite impacts, has been recurrently applied to multiple extinction boundaries. However, these multidisciplinary research efforts across the globe have been largely unsuccessful to date, with one outstanding exception: the Cretaceous-Paleogene boundary. The unicausal impact scenario as a leading explanation, when applied to the complex fossil record, has resulted in force-fitting of data and interpretations ("great expectations syndrome". The misunderstandings can be grouped at three successive levels of the testing process, and involve the unreflective application of the impact paradigm: (i) factual misidentification, i.e., an erroneous or indefinite recognition of the extraterrestrial record in sedimentological, physical and geochemical contexts, (ii) correlative misinterpretation of the adequately documented impact signals due to their incorrect dating, and (iii) causal overestimation when the proved impact characteristics are doubtful as a sufficient trigger of a contemporaneous global cosmic catastrophe. Examples of uncritical belief in the simple cause-effect scenario for the Frasnian-Famennian, Permian-Triassic, and Triassic-Jurassic (and the Eifelian-Givetian and Paleocene-Eocene as well) global events include mostly item-1 pitfalls (factual misidentification), with Ir enrichments and shocked minerals frequently misidentified. Therefore, these mass extinctions are still at the first test level, and only the F-F extinction is potentially seen in the context of item-2, the interpretative step, because of the possible causative link with the Siljan Ring crater (53 km in diameter). The erratically recognized cratering signature is often marked by large timing and size uncertainties, and item-3, the advanced causal inference, is in fact limited to clustered impacts that clearly predate major mass extinctions. The multi-impact lag-time pattern is particularly clear in the Late Triassic, when the largest (100 km diameter) Manicouagan crater was possibly concurrent with the end-Carnian extinction (or with the late Norian tetrapod turnover on an alternative time scale). The relatively small crater sizes and cratonic (crystalline rock basement) setting of these two craters further suggest the strongly insufficient extraterrestrial trigger of worldwide environmental traumas. However, to discuss the kill potential of impact events in a more robust fashion, their location and timing, vulnerability factors, especially target geology and palaeogeography in the context of associated climate-active volatile fluxes, should to be rigorously assessed. The current lack of conclusive impact evidence synchronous with most mass extinctions may still be somewhat misleading due to the predicted large set of undiscovered craters, particularly in light of the obscured record of oceanic impact events

Dimensions of invasiveness: Links between local abundance, geographic range size, and habitat breadth in Europe's alien and native floras

Understanding drivers of success for alien species can inform on potential future invasions. Recent conceptual advances highlight that species may achieve invasiveness via performance along at least three distinct dimensions: 1) local abundance, 2) geographic range size, and 3) habitat breadth in naturalized distributions. Associations among these dimensions and the factors that determine success in each have yet to be assessed at large geographic scales. Here, we combine data from over one million vegetation plots covering the extent of Europe and its habitat diversity with databases on species' distributions, traits, and historical origins to provide a comprehensive assessment of invasiveness dimensions for the European alien seed plant flora. Invasiveness dimensions are linked in alien distributions, leading to a continuum from overall poor invaders to super invaders - abundant, widespread aliens that invade diverse habitats. This pattern echoes relationships among analogous dimensions measured for native European species. Success along invasiveness dimensions was associated with details of alien species' introduction histories: earlier introduction dates were positively associated with all three dimensions, and consistent with theory-based expectations, species originating from other continents, particularly acquisitive growth strategists, were among the most successful invaders in Europe. Despite general correlations among invasiveness dimensions, we identified habitats and traits associated with atypical patterns of success in only one or two dimensions - for example, the role of disturbed habitats in facilitating widespread specialists. We conclude that considering invasiveness within a multidimensional framework can provide insights into invasion processes while also informing general understanding of the dynamics of species distributions.Deutsche Forschungsgemeinschaft (264740629) Grantová Agentura České Republiky (19-28491X) Grantová Agentura České Republiky (19-28807X) Grantová Agentura České Republiky (RVO 67985939) Austrian Science Fund (I 2086 - B29) Bundesministerium für Bildung und Forschung (01LC1807A) Eusko Jaurlaritza (IT299-10) National Research Foundation of Korea (2018R1C1B6005351) University of Latvia (AAp2016/B041//Zd2016/AZ03) Villum Fonden (16549

TRY plant trait database – enhanced coverage and open access

Plant traits—the morphological, anatomical, physiological, biochemical and phenological characteristics of plants—determine how plants respond to environmental factors, affect other trophic levels, and influence ecosystem properties and their benefits and detriments to people. Plant trait data thus represent the basis for a vast area of research spanning from evolutionary biology, community and functional ecology, to biodiversity conservation, ecosystem and landscape management, restoration, biogeography and earth system modelling. Since its foundation in 2007, the TRY database of plant traits has grown continuously. It now provides unprecedented data coverage under an open access data policy and is the main plant trait database used by the research community worldwide. Increasingly, the TRY database also supports new frontiers of trait‐based plant research, including the identification of data gaps and the subsequent mobilization or measurement of new data. To support this development, in this article we evaluate the extent of the trait data compiled in TRY and analyse emerging patterns of data coverage and representativeness. Best species coverage is achieved for categorical traits—almost complete coverage for ‘plant growth form’. However, most traits relevant for ecology and vegetation modelling are characterized by continuous intraspecific variation and trait–environmental relationships. These traits have to be measured on individual plants in their respective environment. Despite unprecedented data coverage, we observe a humbling lack of completeness and representativeness of these continuous traits in many aspects. We, therefore, conclude that reducing data gaps and biases in the TRY database remains a key challenge and requires a coordinated approach to data mobilization and trait measurements. This can only be achieved in collaboration with other initiatives