Moderate deviations for the eigenvalue counting function of Wigner matrices

We establish a moderate deviation principle (MDP) for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on fine asymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson. The extension to large families of Wigner matrices is based on the Tao and Vu Four Moment Theorem and applies localization results by Erd\"os, Yau and Yin. Moreover we investigate families of covariance matrices as well.Comment: 20 page

Connection times in large ad hoc mobile networks

We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances $\leq 2R$, with $2R$ the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random, and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model (RWP). Here we describe the decay rate, in the limit of large time horizons, of the probability that the portion of the connection time is less than the expectation

Connection times in large ad hoc mobile networks

We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances 2R, with 2R the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random, and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model (RWP). Here we describe the decay rate, in the limit of large time horizons, of the probability that the portion of the connection time is less than the expectation

Connection times in large ad-hoc mobile networks

We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, that is, they are iteratively forwarded from participant to participant over distances smaller than the communication radius until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model. Here, we give a negative upper bound for the decay rate, in the limit of large time horizons, of the probability of the event that the portion of the connection time is less than the expectation.Comment: Published at http://dx.doi.org/10.3150/15-BEJ724 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Optically trapped atom interferometry using the clock transition of large Rb-87 Bose-Einstein condensates

We present a Ramsey-type atom interferometer operating with an optically trapped sample of 10^6 Bose-condensed Rb-87 atoms. The optical trap allows us to couple the |F =1, mF =0>\rightarrow |F =2, mF =0> clock states using a single photon 6.8GHz microwave transition, while state selective readout is achieved with absorption imaging. Interference fringes with contrast approaching 100% are observed for short evolution times. We analyse the process of absorption imaging and show that it is possible to observe atom number variance directly, with a signal-to-noise ratio ten times better than the atomic projection noise limit on 10^6 condensate atoms. We discuss the technical and fundamental noise sources that limit our current system, and outline the improvements that can be made. Our results indicate that, with further experimental refinements, it will be possible to produce and measure the output of a sub-shot-noise limited, large atom number BEC-based interferometer. In an addendum to the original paper, we attribute our inability to observe quantum projection noise to the stability of our microwave oscillator and background magnetic field. Numerical simulations of the Gross-Pitaevskii equations for our system show that dephasing due to spatial dynamics driven by interparticle interactions account for much of the observed decay in fringe visibility at long interrogation times. The simulations show good agreement with the experimental data when additional technical decoherence is accounted for, and suggest that the clock states are indeed immiscible. With smaller samples of 5 \times 10^4 atoms, we observe a coherence time of {\tau} = (1.0+0.5-0.3) s.Comment: 22 pages, 6 figures Addendum: 11 pages, 6 figure