Limit theory for the Gilbert graph

For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using a concentration inequality for the convex distance

The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let $\eta_t$ be a Poisson point process of intensity $t\geq 1$ on some state space \Y and $f$ be a non-negative symmetric function on \Y^k for some $k\geq 1$. Applying $f$ to all $k$-tuples of distinct points of $\eta_t$ generates a point process $\xi_t$ on the positive real-half axis. The scaling limit of $\xi_t$ as $t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the the $m$-th smallest point of $\xi_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener-It\^o chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen-Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as Poisson $k$-flats, Poisson random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry

On a CFT limit of planar γi\gamma_i-deformed N=4\mathcal{N}=4 SYM theory

We show that an integrable four-dimensional non-unitary field theory that was recently proposed as a certain limit of the $\gamma_i$-deformed $\mathcal{N}=4$ SYM theory is incomplete and not conformal -- not even in the planar limit. We complete this theory by double-trace couplings and find conformal one-loop fix-points when admitting respective complex coupling constants. These couplings must not be neglected in the planar limit, as they can contribute to planar multi-point functions. Based on our results for certain two-loop planar anomalous dimensions, we propose tests of integrability.Comment: LaTeX, 3 pages, 1 Figur

Central limit theorems for the radial spanning tree

Consider a homogeneous Poisson point process in a compact convex set in $d$-dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length

Fleet management in free-floating bike sharing systems using predictive modelling and explorative tools

For redistribution and operating bikes in a free-floating systems, two measures are of highest priority. First, the information about the expected number of rentals on a day is an important measure for service providers for management and service of their fleet. The estimation of the expected number of bookings is carried out with a simple model and a more complex model based on meterological information, as the number of loans depends strongly on the current and forecasted weather. Secondly, the knowledge of a service level violation in future on a fine spatial resolution is important for redistribution of bikes. With this information, the service provider can set reward zones where service level violations will occur in the near future. To forecast a service level violation on a fine geographical resolution the current distribution of bikes as well as the time and space information of past rentals has to be taken into account. A Markov Chain Model is formulated to integrate this information. We develop a management tool that describes in an explorative way important information about past, present and predicted future counts on rentals in time and space. It integrates all estimation procedures. The management tool is running in the browser and continuously updates the information and predictions since the bike distribution over the observed area is in continous flow as well as new data are generated continuously

Demographics and Volatile Social Security Wealth: Political Risks of Benefit Rule Changes in Germany

In this paper we address the question how the generosity of the benefit rule of the German public pension system has changed during the past three decades and how this development can be explained by demographic changes. Firstly, we illustrate the political risk of benefit rule changes for individuals. We find that depending on the birth year and the considered scenario the relative losses vary between 30 and nearly 60 percent. Secondly, we estimate how demographic developments have triggered these changes in generosity. Our results suggest that future developments of the old-age dependency ratio have an influence on the determination of generosity.social security wealth, demography, political economy, Germany

Impact of Warped Extra Dimensions on the Dipole Coefficients in b→sγ\boldsymbol{b\to s\gamma} Transitions

We calculate the electro- and chromomagnetic dipole coefficients $C_{7\gamma,8g}$ and $\tilde C_{7\gamma,8g}$ in the context of the minimal Randall-Sundrum (RS) model with a Higgs sector localized on the IR brane using the five-dimensional (5D) approach, where the coefficients are expressed in terms of integrals over 5D propagators. Since we keep the full dependence on the Yukawa matrices, the integral expressions are formally valid to all orders in $v^2/M_{\rm KK}^2$. In addition we relate our results to the expressions obtained in the Kaluza-Klein (KK) decomposed theory and show the consistency in both pictures analytically and numerically, which presents a non-trivial cross-check. In Feynman-'t Hooft gauge, the dominant corrections from virtual KK modes arise from the scalar parts of the $W^\pm$-boson penguin diagrams, including the contributions from the scalar component of the 5D gauge-boson field and from the charged Goldstone bosons in the Higgs sector. The size of the KK corrections depends on the parameter $y_\ast$, which sets the upper bound for the anarchic 5D Yukawa matrices. We find that for $y_\ast\gtrsim1$ the dominant KK corrections are proportional to $y_\ast^2$. We discuss the phenomenological implications of our results for the branching ratio ${\rm Br}(\bar B\to X_s\gamma)$, the time-dependent CP asymmetry $S_{K^*\gamma}$, the direct CP asymmetry $A_{\rm CP}^{b\to s\gamma}$ and the CP asymmetry difference $\Delta A_{\rm CP}^{b\to s\gamma}$. We can derive a lower bound on the first KK gluon resonance of $3.8\,$TeV for $y_\ast=3$, requiring that at least $10\%$ of the RS parameter space covers the experimental $2\sigma$ error margins. We further discuss the branching ratio ${\rm Br}(\bar B\to X_sl^+l^-)$ and compare our predictions for $C_{7\gamma,9,10}$ and $\tilde C_{7\gamma,9,10}$ with phenomenological results derived from model-independent analyses.Comment: 44 pages plus appendix, 10 figures, added equations (58) and (61

Higgs Couplings and Phenomenology in a Warped Extra Dimension

We present a comprehensive description of the Higgs-boson couplings to Standard Model fermions and bosons in Randall-Sundrum (RS) models with a Higgs sector localized on or near the infra-red brane. The analytic results for all relevant Higgs couplings including the loop-induced couplings to gluons and photons are summarized for both the minimal and the custodial RS model. The RS predictions for all relevant Higgs decays are compared with current LHC data, which already exclude significant portions of the parameter space. We show that the latest measurements are sensitive to KK gluon masses up to $20\, \rm{TeV} \times (y_*/3)$ at $95\%$ confidence level for anarchic 5D Yukawa couplings bounded from above by $|(Y_f)_{ij}| < y_*$. We also derive the sensitivity levels attainable in the high-luminosity run of the LHC and at a future linear collider.Comment: 28 pages plus appendix, 9 figures; equation (52) corrected, references update