Semisimple Lie groups satisfy property RD, a short proof

We give a short elementary proof of the fact that connected semisimple real Lie groups satisfy property RD. The proof is based on a process of linearization

Complements of hyperplane sub-bundles in projective space bundles over the projective line

We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub-bundle H of a projective space bundle of rank r-1 over the projective line depends only on the the r-fold self-intersection of H . In particular it depends neither on the ambient bundle nor on a particular ample hyperplane sub-bundle with given r-fold self-intersection. Our proof exploits the unexpected property that every such complement comes equipped with the structure of a non trivial torsor under a vector bundle on the affine line with a double origin

The component sizes of a critical random graph with given degree sequence

Consider a critical random multigraph $\mathcal{G}_n$ with $n$ vertices constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution $\nu$ (criticality means that the second moment of $\nu$ is finite and equals twice its first moment). We specify the scaling limits of the ordered sequence of component sizes of $\mathcal{G}_n$ as $n$ tends to infinity in different cases. When $\nu$ has finite third moment, the components sizes rescaled by $n^{-2/3}$ converge to the excursion lengths of a Brownian motion with parabolic drift above past minima, whereas when $\nu$ is a power law distribution with exponent $\gamma\in(3,4)$, the components sizes rescaled by $n^{-(\gamma -2)/(\gamma-1)}$ converge to the excursion lengths of a certain nontrivial drifted process with independent increments above past minima. We deduce the asymptotic behavior of the component sizes of a critical random simple graph when $\nu$ has finite third moment.Comment: Published in at http://dx.doi.org/10.1214/13-AAP985 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org