Mathematics of random growing interfaces

We establish a thermodynamic limit and Gaussian fluctuations for the height and surface width of the random interface formed by the deposition of particles on surfaces. The results hold for the standard ballistic deposition model as well as the surface relaxation model in the off-lattice setting. The results are proved with the aid of general limit theorems for stabilizing functionals of marked Poisson point processes.Comment: 12 page

Degree Correlations in Random Geometric Graphs

Spatially embedded networks are important in several disciplines. The prototypical spatial net- work we assume is the Random Geometric Graph of which many properties are known. Here we present new results for the two-point degree correlation function in terms of the clustering coefficient of the graphs for two-dimensional space in particular, with extensions to arbitrary finite dimension

Weak Cosmic Censorship: As Strong as Ever

Spacetime singularities that arise in gravitational collapse are always hidden inside of black holes. This is the essence of the weak cosmic censorship conjecture. The hypothesis, put forward by Penrose 40 years ago, is still one of the most important open questions in general relativity. In this Letter, we reanalyze extreme situations which have been considered as counterexamples to the weak cosmic censorship conjecture. In particular, we consider the absorption of scalar particles with large angular momentum by a black hole. Ignoring back reaction effects may lead one to conclude that the incident wave may overspin the black hole, thereby exposing its inner singularity to distant observers. However, we show that when back reaction effects are properly taken into account, the stability of the black-hole event horizon is irrefutable. We therefore conclude that cosmic censorship is actually respected in this type of gedanken experiments.Comment: 4 page

Twistors, special relativity, conformal symmetry and minimal coupling - a review

An approach to special relativistic dynamics using the language of spinors and twistors is presented. Exploiting the natural conformally invariant symplectic structure of the twistor space, a model is constructed which describes a relativistic massive, spinning and charged particle, minimally coupled to an external electro-magnetic field. On the two-twistor phase space the relativistic Hamiltonian dynamics is generated by a Poincare scalar function obtained from the classical limit (appropriately defined by us) of the second order, to an external electro-magnetic field minimally coupled, Dirac operator. In the so defined relativistic classical limit there are no Grassman variables. Besides, the arising equation that describes dynamics of the relativistic spin differs significantly from the so called Thomas Bergman Michel Telegdi equation.Comment: 39 pages, no figures, few erronous statements (not affecting anything else in the papper) on page 23 delete

Integrable relativistic systems given by Hamiltonians with momentum-spin-orbit coupling

In the paper we investigate the evolution of the relativistic particle (massive and massless) with spin defined by Hamiltonian containing the terms with momentum-spin-orbit coupling. We integrate the corresponding Hamiltonian equations in quadratures and express their solutions in terms of elliptic functions.Comment: 18 page

Strict inequalities of critical values in continuum percolation

We consider the supercritical finite-range random connection model where the points $x,y$ of a homogeneous planar Poisson process are connected with probability $f(|y-x|)$ for a given $f$. Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality $p_c^{\rm site} > p_c^{\rm bond}$. We also show that reducing the connection function $f$ strictly increases the critical Poisson intensity. Finally, we deduce that performing a spreading transformation on $f$ (thereby allowing connections over greater distances but with lower probabilities, leaving average degrees unchanged) {\em strictly} reduces the critical Poisson intensity. This is of practical relevance, indicating that in many real networks it is in principle possible to exploit the presence of spread-out, long range connections, to achieve connectivity at a strictly lower density value.Comment: 38 pages, 8 figure

Spinor calculus on 5-dimensional spacetimes

Penrose's spinor calculus of 4-dimensional Lorentzian geometry is extended to the case of 5-dimensional Lorentzian geometry. Such fruitful ideas in Penrose's spinor calculus as the spin covariant derivative, the curvature spinors or the definition of the spin coefficients on a spin frame can be carried over to the spinor calculus in 5-dimensional Lorentzian geometry. The algebraic and differential properties of the curvature spinors are studied in detail and as an application we extend the well-known 4-dimensional Newman-Penrose formalism to a 5-dimensional spacetime.Comment: Convention mismatch and minor typos fixed. To appear in Journal of Mathematical Physic

Positive cosmological constant in loop quantum cosmology

The k=0 Friedmann Lemaitre Robertson Walker model with a positive cosmological constant and a massless scalar field is analyzed in detail. If one uses the scalar field as relational time, new features arise already in the Hamiltonian framework of classical general relativity: In a finite interval of relational time, the universe expands out to infinite proper time and zero matter density. In the deparameterized quantum theory, the true Hamiltonian now fails to be essentially self-adjoint both in the Wheeler DeWitt (WDW) approach and in LQC. Irrespective of the choice of the self-adjoint extension, the big bang singularity persists in the WDW theory while it is resolved and replaced by a big bounce in loop quantum cosmology (LQC). Furthermore, the quantum evolution is surprisingly insensitive to the choice of the self-adjoint extension. This may be a special case of an yet to be discovered general property of a certain class of symmetric operators that fail to be essentially self-adjoint.Comment: 36 pages, 6 figures, RevTex

Approximate twistors and positive mass

In this paper the problem of comparing initial data to a reference solution for the vacuum Einstein field equations is considered. This is not done in a coordinate sense, but through quantification of the deviation from a specific symmetry. In a recent paper [T. B\"ackdahl, J.A. Valiente Kroon, Phys. Rev. Lett. 104, 231102 (2010)] this problem was studied with the Kerr solution as a reference solution. This analysis was based on valence 2 Killing spinors. In order to better understand this construction, in the present article we analyse the analogous construction for valence 1 spinors solving the twistor equation. This yields an invariant that measures how much the initial data deviates from Minkowski data. Furthermore, we prove that this invariant vanishes if and only of the mass vanishes. Hence, we get a proof of the positivity of mass.Comment: 18 pages, corrected typos, updated reference

The Measure Problem in Cosmology

The Hamiltonian structure of general relativity provides a natural canonical measure on the space of all classical universes, i.e., the multiverse. We review this construction and show how one can visualize the measure in terms of a "magnetic flux" of solutions through phase space. Previous studies identified a divergence in the measure, which we observe to be due to the dilatation invariance of flat FRW universes. We show that the divergence is removed if we identify universes which are so flat they cannot be observationally distinguished. The resulting measure is independent of time and of the choice of coordinates on the space of fields. We further show that, for some quantities of interest, the measure is very insensitive to the details of how the identification is made. One such quantity is the probability of inflation in simple scalar field models. We find that, according to our implementation of the canonical measure, the probability for N e-folds of inflation in single-field, slow-roll models is suppressed by of order exp(-3N) and we discuss the implications of this result.Comment: 22 pages, 6 figures. Revised version with clarifying remarks on meaning of adopted measure, extra references and minor typographical correction