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Article published in the Comm. Law.

A refined factorization of the exponential law

Let $\xi$ be a (possibly killed) subordinator with Laplace exponent $\phi$ and denote by $I_{\phi}=\int_0^{\infty}\mathrm{e}^{-\xi_s}\,\mathrm{d}s$, the so-called exponential functional. Consider the positive random variable $I_{\psi_1}$ whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106], is determined by its negative entire moments as follows: $$\mathbb {E}[I_{\psi_1}^{-n}]=\prod_{k=1}^n\phi(k),\qquad n=1,2,...$$ In this note, we show that $I_{\psi_1}$ is a positive self-decomposable random variable whenever the L\'{e}vy measure of $\xi$ is absolutely continuous with a monotone decreasing density. In fact, $I_{\psi_1}$ is identified as the exponential functional of a spectrally negative (sn, for short) L\'{e}vy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106] the following factorization of the exponential law ${\mathbf {e}}$: $$I_{\phi}/I_{\psi_1}\stackrel{\mathrm {(d)}}{=}{\mathbf {e}},$$ where $I_{\psi_1}$ is taken to be independent of $I_{\phi}$. We proceed by showing an identity in distribution between the entrance law of an sn self-similar positive Feller process and the reciprocal of the exponential functional of sn L\'{e}vy processes. As a by-product, we obtain some new examples of the law of the exponential functionals, a new factorization of the exponential law and some interesting distributional properties of some random variables. For instance, we obtain that $S(\alpha)^{\alpha}$ is a self-decomposable random variable, where $S(\alpha)$ is a positive stable random variable of index $\alpha\in(0,1)$.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ292 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On the 2-categories of weak distributive laws

A weak mixed distributive law (also called weak entwining structure) in a 2-category consists of a monad and a comonad, together with a 2-cell relating them in a way which generalizes a mixed distributive law due to Beck. We show that a weak mixed distributive law can be described as a compatible pair of a monad and a comonad, in 2-categories extending, respectively, the 2-category of comonads and the 2-category of monads. Based on this observation, we define a 2-category whose 0-cells are weak mixed distributive laws. In a 2-category K which admits Eilenberg-Moore constructions both for monads and comonads, and in which idempotent 2-cells split, we construct a fully faithful 2-functor from this 2-category of weak mixed distributive laws to K^{2 x 2}.Comment: 15 pages LaTeX source, final version to appear in Comm. Algebr

Spin liquid behaviour in Jeff=1/2 triangular lattice Ba3IrTi2O9

Ba3IrTi2O9 crystallizes in a hexagonal structure consisting of a layered triangular arrangement of Ir4+ (Jeff=1/2). Magnetic susceptibility and heat capacity data show no magnetic ordering down to 0.35K inspite of a strong magnetic coupling as evidenced by a large Curie-Weiss temperature=-130K. The magnetic heat capacity follows a power law at low temperature. Our measurements suggest that Ba3IrTi2O9 is a 5d, Ir-based (Jeff=1/2), quantum spin liquid on a 2D triangular lattice.Comment: 10 pages including supplemental material, to be published in Phys. Rev. B (Rapid Comm.

Emergence of L\'{e}vy Walks in Systems of Interacting Individuals

Recent experiments (G. Ariel, et al., Nature Comm. 6, 8396 (2015)) revealed an intriguing behavior of swarming bacteria: they fundamentally change their collective motion from simple diffusion into a superdiffusive L\'{e}vy walk dynamics. We introduce a nonlinear non-Markovian persistent random walk model that explains the emergence of superdiffusive L\'{e}vy walks. We show that the alignment interaction between individuals can lead to the superdiffusive growth of the mean squared displacement and the power law distribution of run length with infinite variance. The main result is that the superdiffusive behavior emerges as a nonlinear collective phenomenon, rather than due to the standard assumption of the power law distribution of run distances from the inception. At the same time, we find that the repulsion/collision effects lead to the density dependent exponential tempering of power law distributions. This qualitatively explains experimentally observed transition from superdiffusion to the diffusion of mussels as their density increases (M. de Jager et al., Proc. R. Soc. B 281, 20132605 (2014))

Scaling law and stability for a noisy quantum system

We show that a scaling law exists for the near resonant dynamics of cold kicked atoms in the presence of a randomly fluctuating pulse amplitude. Analysis of a quasi-classical phase-space representation of the quantum system with noise allows a new scaling law to be deduced. The scaling law and associated stability are confirmed by comparison with quantum simulations and experimental data.Comment: Published in Physical Review E (Rapid Comm.

Random matrices: Universality of local eigenvalue statistics up to the edge

This is a continuation of our earlier paper on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in that paper from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.Comment: 24 pages, no figures, to appear, Comm. Math. Phys. One new reference adde