Exponential tail bounds for loop-erased random walk in two dimensions

Let $M_n$ be the number of steps of the loop-erasure of a simple random walk on $\mathbb{Z}^2$ from the origin to the circle of radius $n$. We relate the moments of $M_n$ to $Es(n)$, the probability that a random walk and an independent loop-erased random walk both started at the origin do not intersect up to leaving the ball of radius $n$. This allows us to show that there exists $C$ such that for all $n$ and all $k=1,2,...,\mathbf{E}[M_n^k]\leq C^kk!\mathbf{E}[M_n]^k$ and hence to establish exponential moment bounds for $M_n$. This implies that there exists $c>0$ such that for all $n$ and all $\lambda\geq0$, $$\mathbf{P}\{M_n>\lambda\mathbf{E}[M_n]\}\leq2e^{-c\lambda}.$$ Using similar techniques, we then establish a second moment result for a specific conditioned random walk which enables us to prove that for any $\alpha0$ such that for all $n$ and $\lambda>0$, $$\mathbf{P}\{M_n<\lambda^{-1}\mathbf{E}[M_n]\}\leq Ce^{-c'\lambda ^{\alpha}}.$$Comment: Published in at http://dx.doi.org/10.1214/10-AOP539 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Noncommutative generalization of SU(n)-principal fiber bundles: a review

This is an extended version of a communication made at the international conference ``Noncommutative Geometry and Physics'' held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary fiber bundle theory. The noncommutative algebra is the endomorphism algebra of a SU(n)-vector bundle, and its differential calculus is based on its Lie algebra of derivations. It is shown that this noncommutative geometry contains some of the most important constructions introduced and used in the theory of connections on vector bundles, in particular, what is needed to introduce gauge models in physics, and it also contains naturally the essential aspects of the Higgs fields and its associated mechanics of mass generation. It permits one also to extend some previous constructions, as for instance symmetric reduction of (here noncommutative) connections. From a mathematical point of view, these geometrico-algebraic considerations highlight some new point on view, in particular we introduce a new construction of the Chern characteristic classes

κ\kappa-deformation, affine group and spectral triples

A regular spectral triple is proposed for a two-dimensional $\kappa$-deformation. It is based on the naturally associated affine group $G$, a smooth subalgebra of $C^*(G)$, and an operator \caD defined by two derivations on this subalgebra. While \caD has metric dimension two, the spectral dimension of the triple is one. This bypasses an obstruction described in \cite{IochMassSchu11a} on existence of finitely-summable spectral triples for a compactified $\kappa$-deformation.Comment: 29 page

Linear Connections on the Two Parameter Quantum Plane

We apply a recently proposed definition of a linear connection in non commutative geometry based on the natural bimodule structure of the algebra of differential forms to the case of the two-parameter quantum plane. We find that there exists a non trivial family of linear connections only when the two parameters obeys a specific relation.Comment: 7 pages, Te

Surface figure measurements of radio telescopes with a shearing interferometer

A new technique for determining the surface figure of large submillimeter wavelength telescopes is presented, which is based on measuring the telescope’s focal plane diffraction pattern with a shearing interferometer. In addition to the instrumental theory, results obtained using such an interferometer on the 10.4-m diam telescope of the Caltech Submillimeter Observatory are discussed. Using wavelengths near 1 mm, a measurement accuracy of 9 µm, or λ/115, has been achieved, and the rms surface accuracy has been determined to be just under 30 µm. The distortions of the primary reflector with changing elevation angle have also been measured and agree well with theoretical predictions of the dish deformation

Belief Hierarchical Clustering

In the data mining field many clustering methods have been proposed, yet standard versions do not take into account uncertain databases. This paper deals with a new approach to cluster uncertain data by using a hierarchical clustering defined within the belief function framework. The main objective of the belief hierarchical clustering is to allow an object to belong to one or several clusters. To each belonging, a degree of belief is associated, and clusters are combined based on the pignistic properties. Experiments with real uncertain data show that our proposed method can be considered as a propitious tool

On the Interpretation of the broad-band millimeter-wave flux from Orion

Spectral observations of the core of Orion A at wavelengths around 1.3 mm show a high density of strong, broad emission lines. The combined flux in lines with peak antenna temperatures stronger than 0.2 K accounts for approximately 40 percent of the broad-band millimeter-wave flux from the region. Thus the broad-band flux from Orion A is in large part due to sources other than dust emission

Molecular abundances in OMC-1: The chemical composition of interstellar molecular clouds and the influence of massive star formation

We present here an investigation of the chemical composition of the various regions in the core of the Orion molecular cloud (OMC-1) based on results from the Caltech Owens Valley Radio Observatory (OVRO) millimeter-wave spectral line survey (Sutton et al.; Blake et al.). This survey covered a 55 GHz interval in the 1.3 mm (230 GHz) atmospheric window and contained emission from over 800 resolved spectral features. Of the 29 identified species 14 have a sufficient number of detected transitions to be investigated with an LTE "rotation diagram" technique, in which large numbers of lines are used to estimate both the rotational excitation and the overall abundance. The rotational temperatures and column densities resulting from these fits have then been used to model the emission from those remaining species which either have too few lines or which are too weak to be so analyzed. When different kinematic sources of emission are blended to produce a single feature, Gaussian fits have been used to derive the individual contributions to the total line profile. The uniformly calibrated data in the unique and extensive Caltech spectral line survey lead to accurate estimates of the chemical and physical parameters of the Orion molecular cloud, and place significant constraints on models of interstellar chemistry. A global analysis of the observed abundances shows that the markedly different chemical compositions of the kinematically and spatially distinct Orion subsources may be interpreted in the framework of an evolving, initially quiescent, gas-phase chemistry influenced by the process of massive star formation. The chemical composition of the extended Orion cloud complex is similar to that found in a number of other objects, but the central regions of OMC-1 have had their chemistry selectively altered by the radiation and high-velocity outflow from the young stars embedded deep within the interior of the molecular cloud. Specifically, the extended ridge clouds are inferred to have a low (subsolar) gas-phase oxygen content from the prevalence of reactive carbon-rich species like CN, CCH, and C_3H_2 also found in more truly quiescent objects such as TMC-1. The similar abundances of these and other simple species in clouds like OMC-1, Sgr B2, and TMC-1 lend support to gas-phase ion-molecule models of interstellar chemistry, but grain processes may also play a significant role in maintaining the overall chemical balance in such regions through selective depletion mechanisms and grain mantle processing. In contrast, the chemical compositions of the more turbulent plateau and hot core components of OMC-1 are dominated by high-temperature, shock-induced gas and grain surface neutral-neutral reaction processes. The high silicon/sulfur oxide and water content of the plateau gas is best modeled by fast shock disruption of smaller grain cores to release the more refractory elements followed by a predominantly neutral chemistry in the cooling postshock regions, while a more passive release of grain mantle products driven toward kinetic equilibrium most naturally explains the prominence of fully hydrogenated N-containing species like HCN, NH_3 , CH_3CN, and C_2H_5CN in the hot core. The clumpy nature of the outflow is illustrated by the high-velocity emission observed from easily decomposed molecules such as H_2CO. Areas immediately adjacent to the shocked core in which the cooler, ion-rich gas of the surrounding molecular cloud is mixed with water/oxygen rich gas from the plateau source are proposed to give rise to the enhanced abundances of complex internal rotors such as CH_30H, HCOOCH_3 , and CH_30CH_3 whose line widths are similar to carbon-rich species such as CN and CCH found in the extended ridge, but whose rotational temperatures are somewhat higher and whose spatial extents are much more compact