On Finite Rank Deformations of Wigner Matrices II: Delocalized Perturbations

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices. We assume that the matrix entries have finite fourth moment and extend the results by Capitaine, Donati-Martin, and F\'eral for perturbations whose eigenvectors are delocalized.Comment: We explained some proofs in greater detail, corrected several small misprints, and updated the bibliograph

Eigenvalues of block structured asymmetric random matrices

We study the spectrum of an asymmetric random matrix with block structured variances. The rows and columns of the random square matrix are divided into $D$ partitions with arbitrary size (linear in $N$). The parameters of the model are the variances of elements in each block, summarized in $g\in\mathbb{R}^{D\times D}_+$. Using the Hermitization approach and by studying the matrix-valued Stieltjes transform we show that these matrices have a circularly symmetric spectrum, we give an explicit formula for their spectral radius and a set of implicit equations for the full density function. We discuss applications of this model to neural networks

Power law decay for systems of randomly coupled differential equations

We consider large random matrices $X$ with centered, independent entries but possibly different variances. We compute the normalized trace of $f(X) g(X^*)$ for $f,g$ functions analytic on the spectrum of $X$. We use these results to compute the long time asymptotics for systems of coupled differential equations with random coefficients. We show that when the coupling is critical the norm squared of the solution decays like $t^{-1/2}$.Comment: 20 pages, Corrected a typo in Assumption (1) [after final publication] and made other irrelevant revision

On Finite Rank Deformations of Wigner Matrices

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrice under the assumption that the off-diagonal matrix entries have uniformly bounded fifth moment and the diagonal entries have uniformly bounded third moment. Using our recent results on the fluctuation of resolvent entries [31],[28], and ideas from [9], we extend results by M.Capitaine, C.Donati-Martin, and D.F\'eral [12], [13].Comment: accepted for publication in Annales de l'Institut Henri Poincar\'e (B) Probabilit\'es et Statistique

TRAFFIC SIGNAL CONTROL WITH ANT COLONY OPTIMIZATION

Traffic signal control is an effective way to improve the efficiency of traffic networks and reduce users’ delays. Ant Colony Optimization (ACO) is a metaheuristic based on the behavior of ant colonies searching for food. ACO has successfully been used to solve many NP-hard combinatorial optimization problems and its stochastic and decentralized nature fits well with traffic flow networks. This thesis investigates the application of ACO to minimize user delay at traffic intersections. Computer simulation results show that this new approach outperforms conventional fully actuated control under the condition of high traffic demand

A parameterization of Greenland's tip jets suitable for ocean or coupled climate models

Greenland's tip jets are low-level, high wind speed jets forced by an interaction of the synoptic-scale atmospheric flow and the steep, high orography of Greenland. These jets are thought to play an important role in both preconditioning for, and triggering of, open-ocean convection in the Irminger Sea. However, the relatively small spatial scale of the jets prevents their accurate representation in the relatively low resolution (~1 degree) atmospheric (re-)analyses which are typically used to force ocean general circulation models (e.g. ECMWF ERA-40 and NCEP reanalyses, or products based on these). Here we present a method of ‘bogussing’ Greenland's tip jets into such surface wind fields and thus, via bulk flux formulae, into the air-sea turbulent flux fields. In this way the full impact of these mesoscale tip jets can be incorporated in any ocean general circulation model of sufficient resolution. The tip jet parameterization is relatively simple, making use of observed linear gradients in wind speed along and across the jet, but is shown to be accurate to a few m s-1 on average. The inclusion of tip jets results in a large local increase in both the heat and momentum fluxes. When applied to a 1-dimensional mixed-layer model this results in a deepening of the winter mixed-layer of over 300 m. The parameterization scheme only requires 10 meter wind speed and mean sea level pressure as input fields; thus it is also suitable for incorporation into a coupled atmosphere-ocean climate model at the coupling stage

On the low dimensional dynamics of structured random networks

Using a generalized random recurrent neural network model, and by extending our recently developed mean-field approach [J. Aljadeff, M. Stern, T. Sharpee, Phys. Rev. Lett. 114, 088101 (2015)], we study the relationship between the network connectivity structure and its low dimensional dynamics. Each connection in the network is a random number with mean 0 and variance that depends on pre- and post-synaptic neurons through a sufficiently smooth function $g$ of their identities. We find that these networks undergo a phase transition from a silent to a chaotic state at a critical point we derive as a function of $g$. Above the critical point, although unit activation levels are chaotic, their autocorrelation functions are restricted to a low dimensional subspace. This provides a direct link between the network's structure and some of its functional characteristics. We discuss example applications of the general results to neuroscience where we derive the support of the spectrum of connectivity matrices with heterogeneous and possibly correlated degree distributions, and to ecology where we study the stability of the cascade model for food web structure.Comment: 16 pages, 4 figure

Singularity degree of structured random matrices

We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up, which we label the singularity degree