Hanna Döring

Moderate Deviations via Cumulants

The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations probabilities due to Rudzkis, Saulis, and Statulevičius. The examples of random objects we treat include dependency graphs, subgraph-counting statistics in Erdös–Rényi random graphs and U-statistics. Moreover, we prove moderate deviation principles for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices and the number of particles in a growing box of random determinantal point processes such as the number of eigenvalues in the GUE or the number of points in Airy, Bessel, and sine random point fields.

Velocity correlation functions of charged test particles

In recent years it has become clear that a nonlinear formulation of test-particle theories has to be preferred over the traditional quasilinear approach. Nonlinear transport theories such as the nonlinear guiding-centre theory or the weakly nonlinear theory are based on certain assumptions which cannot be derived systematically. One of the key inputs into these theories is the velocity correlation function. In the current paper the Taylor–Green–Kubo formulation is used to deduce a general relation between the mean square displacement of the particle, the running diffusion coefficient and the velocity correlation function. This relation can be used to extract velocity correlation function from test-particle simulations and from transport theories. The latter possibility is the subject of the current paper. These results, which are essential for the improvement of nonlinear transport theories, are compared with standard models applied previously. An additional result of this paper is that for realistic wave spectra, perpendicular diffusion is recovered even within the magnetostatic slab model.

Perpendicular transport of charged particles in slab turbulence: recovery of diffusion for realistic wavespectra?

It is a standard assumption that parallel scattering suppresses perpendicular diffusion to a subdiffusive level when the turbulence lacks transverse structure. In several previous papers, it has been demonstrated that a running or time-dependent diffusion coefficient decreases with increasing time. In most studies, it was assumed that this subdiffusive behaviour is directly related to the slab structure of turbulence. By applying reliable analytical calculations in combination with realistic forms of the wavespectrum of interplanetary turbulence, it is demonstrated in this paper that the long-time behaviour of running diffusion coefficients depends on the wavespectrum in the energy range. If the energy range spectral index exceeds unity, diffusion is recovered. This recovery of diffusion obtained within the magnetostatic slab model is an unexpected result.

Moderate Deviations for the Determinant of Wigner Matrices

We establish a moderate deviations principle (MDP) for the log-determinant log | det(M n ) | of a Wigner matrix M n matching four moments with either the GUE or GOE ensemble. Further we establish Cramer-type moderate deviations and Berry-Esseen bounds for the log-determinant for the GUE and GOE ensembles as well as for non-symmetric and non-Hermitian Gaussian random matrices (Ginibre ensembles), respectively.

Edge Fluctuations of Eigenvalues of Wigner Matrices

We establish a moderate deviation principle (MDP) for the number of eigenvalues of a Wigner matrix in an interval close to the edge of the spectrum. Moreover we prove a MDP for the jth largest eigenvalue close to the edge. 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. Possible extensions to other random matrix ensembles are commented.

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.