Decomposition of spectral density in individual eigenvalue contributions

The eigenvalue densities of two random matrix ensembles, the Wigner Gaussian matrices and the Wishart covariant matrices, are decomposed in the contributions of each individual eigenvalue distribution. It is shown that the fluctuations of all eigenvalues, for medium matrix sizes, are described with a good precision by nearly normal distributions.

We're Chained: an analysis of systemic risk in finance

This dissertation presents two papers on how to deal with simple systemic risk measures to assess portfolio risk characteristics. The first paper deals with the Granger-causation of systemic risk indicators based in correlation matrices in stock returns. Special focus is devoted to the Eigenvalue Entropy as some previous literature indicated strong re- sults, but not considering different macroeconomic scenarios; the Index Cohesion Force and the Absorption Ratio are also considered. Considering the S&P500, there is not ev- idence of Granger-causation from Eigenvalue Entropies and the Index Cohesion Force. The Absorption Ratio Granger-caused both the S&P500 and the VIX index, being the only simple measure that passed this test. The second paper develops this measure to capture the regimes underlying the American stock market. New indicators are built using filtering and random matrix theory. The returns of the S&P500 is modelled as a mixture of normal distributions. The activation of each normal distribution is governed by a Markov chain with the transition probabilities being a function of the indicators. The model shows that using a Herfindahl-Hirschman Index of the normalized eigenval- ues exhibits best fit to the returns from 1998-2013.

Aninhamento em redes bipartidas

We present a nestedness index that measures the nestedness pattern of bipartite networks, a problem that arises in theoretical ecology. Our measure is derived using the sum of distances of the occupied elements in the adjacency matrix of the network. This index quantifies directly the deviation of a given matrix from the nested pattern. In the most simple case the distance of the matrix element ai,j is di,j = i+j, the Manhattan distance. A generic distance is obtained as di,j = (i¬ + j¬)1/¬. The nestedness índex is defined by = 1 − where is the temperature of the matrix. We construct the temperature index using two benchmarks: the distance of the complete nested matrix that corresponds to zero temperature and the distance of the average random matrix that is defined as temperature one. We discuss an important feature of the problem: matrix occupancy. We address this question using a metric index ¬ that adjusts for matrix occupancy

The spectrum of D-slash in QCD via replicas

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Fluctuating commutative geometry

We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number n of points. The spectral principle of Connes and Chamseddine is used to define dynamics. We show that this simple model has two phases. The expectation value , the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension delta is a dynamical observable in our model, and plays the role of an order parameter. The computation of is discussed and an upper bound is found, < 2. We also address another discrete model defined on a fixed d = 1 dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology.

Universalidade em matrizes aleatórias via problemas de Riemann-Hilbert

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Spectral fluctuation and 1/f(alpha) noise in the energy level statistics of interacting trapped bosons

It has been recently shown numerically that the transition from integrability to chaos in quantum systems and the corresponding spectral fluctuations are characterized by 1/f(alpha) noise with 1 <= alpha <= 2. The system of interacting trapped bosons is inhomogeneous and complex. The presence of an external harmonic trap makes it more interesting as, in the atomic trap, the bosons occupy partly degenerate single-particle states. Earlier theoretical and experimental results show that at zero temperature the low-lying levels are of a collective nature and high-lying excitations are of a single-particle nature. We observe that for few bosons, the P(s) distribution shows the Shnirelman peak, which exhibits a large number of quasidegenerate states. For a large number of bosons the low-lying levels are strongly affected by the interatomic interaction, and the corresponding level fluctuation shows a transition to a Wigner distribution with an increase in particle number. It does not follow Gaussian orthogonal ensemble random matrix predictions. For high-lying levels we observe the uncorrelated Poisson distribution. Thus it may be a very realistic system to prove that 1/f(alpha) noise is ubiquitous in nature.

Energy-level statistics of interacting trapped bosons

It is a well-established fact that statistical properties of energy-level spectra are the most efficient tool to characterize nonintegrable quantum systems. The statistical behavior of different systems such as complex atoms, atomic nuclei, two-dimensional Hamiltonians, quantum billiards, and noninteracting many bosons has been studied. The study of statistical properties and spectral fluctuations in interacting many-boson systems has developed interest in this direction. We are especially interested in weakly interacting trapped bosons in the context of Bose-Einstein condensation (BEC) as the energy spectrum shows a transition from a collective nature to a single-particle nature with an increase in the number of levels. However this has received less attention as it is believed that the system may exhibit Poisson-like fluctuations due to the existence of an external harmonic trap. Here we compute numerically the energy levels of the zero-temperature many-boson systems which are weakly interacting through the van der Waals potential and are confined in the three-dimensional harmonic potential. We study the nearest-neighbor spacing distribution and the spectral rigidity by unfolding the spectrum. It is found that an increase in the number of energy levels for repulsive BEC induces a transition from a Wigner-like form displaying level repulsion to the Poisson distribution for P(s). It does not follow the Gaussian orthogonal ensemble prediction. For repulsive interaction, the lower levels are correlated and manifest level-repulsion. For intermediate levels P(s) shows mixed statistics, which clearly signifies the existence of two energy scales: external trap and interatomic interaction, whereas for very high levels the trapping potential dominates, generating a Poisson distribution. Comparison with mean-field results for lower levels are also presented. For attractive BEC near the critical point we observe the Shnirelman-like peak near s = 0, which signifies the presence of a large number of quasidegenerate states.

Quantifying the periodicty of Inter Movement Intervals of Leg Movements

ims: Periodic leg movements in sleep (PLMS) are a frequent finding in polysomnography. Most patients with restless legs syndrome (RLS) display PLMS. However, since PLMS are also often recorded in healthy elderly subjects, the clinical significance of PLMS is still discussed controversially. Leg movements are seen concurrently with arousals in obstructive sleep apnoea (OSA) may also appear periodically. Quantitative assessment of the periodicity of LM/PLM as measured by inter movement intervals (IMI) is difficult. This is mainly due to influencing factors like sleep architecture and sleep stage, medication, inter and intra patient variability, the arbitrary amplitude and sequence criteria which tend to broaden the IMI distributions or make them even multi-modal. Methods: Here a statistical method is presented that enables eliminating such effects from the raw data before analysing the statistics of IMI. Rather than studying the absolute size of IMI (measured in seconds) we focus on the shape of their distribution (suitably normalized IMI). To this end we employ methods developed in Random Matrix Theory (RMT). Patients: The periodicity of leg movements (LM) of four patient groups (10 to 15 each) showing LM without PLMS (group 1), OSA without PLMS (group 2), PLMS and OSA (group 3) as well as PLMS without OSA (group 4) are compared. Results: The IMI of patients without PLMS (groups 1 and 2) and with PLMS (groups 3 and 4) are statistically different. In patients without PLMS the distribution of normalized IMI resembles closely the one of random events. In contrary IMI of PLMS patients show features of periodic systems (e.g. a pendulum) when studied in normalized manner. Conclusions: For quantifying PLMS periodicity proper normalization of the IMI is crucial. Without this procedure important features are hidden when grouping LM/PLM over whole nights or across patients. The clinical significance of PLMS might be eluded when properly separating random LM from LM that show features of periodic systems.

Percolation in soils at Different Bulk Densities: Porosity, Connectivity and Grey Threshold.

To improve percolation modelling on soils the geometrical properties of the pore space must be understood; this includes porosity, particle and pore size distribution and connectivity of the pores. A study was conducted with a soil at different bulk densities based on 3D grey images acquired by X-ray computed tomography. The objective was to analyze the effect in percolation of aspects of pore network geometry and discuss the influence of the grey threshold applied to the images. A model based on random walk algorithms was applied to the images, combining five bulk densities with up to six threshold values per density. This allowed for a dynamical perspective of soil structure in relation to water transport through the inclusion of percolation speed in the analyses. To evaluate separately connectivity and isolate the effect of the grey threshold, a critical value of 35% of porosity was selected for every density. This value was the smallest at which total-percolation walks appeared for the all images of the same porosity and may represent a situation of percolation comparable among bulks densities. This criterion avoided an arbitrary decision in grey thresholds. Besides, a random matrix simulation at 35% of porosity with real images was used to test the existence of pore connectivity as a consequence of a non-random soil structure.

The vibrational energy flow transition in organic molecules: Theory meets experiment

Most large dynamical systems are thought to have ergodic dynamics, whereas small systems may not have free interchange of energy between degrees of freedom. This assumption is made in many areas of chemistry and physics, ranging from nuclei to reacting molecules and on to quantum dots. We examine the transition to facile vibrational energy flow in a large set of organic molecules as molecular size is increased. Both analytical and computational results based on local random matrix models describe the transition to unrestricted vibrational energy flow in these molecules. In particular, the models connect the number of states participating in intramolecular energy flow to simple molecular properties such as the molecular size and the distribution of vibrational frequencies. The transition itself is governed by a local anharmonic coupling strength and a local state density. The theoretical results for the transition characteristics compare well with those implied by experimental measurements using IR fluorescence spectroscopy of dilution factors reported by Stewart and McDonald [Stewart, G. M. & McDonald, J. D. (1983) J. Chem. Phys. 78, 3907–3915].

Testing integrability with a single bit of quantum information

We show that deterministic quantum computing with a single bit can determine whether the classical limit of a quantum system is chaotic or integrable using O(N) physical resources, where N is the dimension of the Hilbert space of the system under study. This is a square-root improvement over all known classical procedures. Our study relies strictly on the random matrix conjecture. We also present numerical results for the nonlinear kicked top.

Marginal Densities of the Wishart Distribution

2010 Mathematics Subject Classification: 62H10.

Approach to chaos in ultracold atomic and molecular physics: Statistics of near-threshold bound states for Li+CaH and Li+CaF

We calculate near-threshold bound states and Feshbach resonance positions for atom–rigid-rotor models of the highly anisotropic systems Li+CaH and Li+CaF. We perform statistical analysis on the resonance positions to compare with the predictions of random matrix theory. For Li+CaH with total angular momentum J=0 we find fully chaotic behavior in both the nearest-neighbor spacing distribution and the level number variance. However, for J>0 we find different behavior due to the presence of a nearly conserved quantum number. Li+CaF (J=0) also shows apparently reduced levels of chaotic behavior despite its stronger effective coupling. This may indicate the development of another good quantum number relating to a bending motion of the complex. However, continuously varying the rotational constant over a wide range shows unexpected structure in the degree of chaotic behavior, including a dramatic reduction around the rotational constant of CaF. This demonstrates the complexity of the relationship between coupling and chaotic behavior.