Thomas Guhr

Random matrix theories in quantum physics: Common concepts

We review the development of random-matrix theory (RMT) during the last fifteen years. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new “statistical mechanics”: Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Identifying States of a Financial Market

The understanding of complex systems has become a central issue because such systems exist in a wide range of scientific disciplines. We here focus on financial markets as an example of a complex system. In particular we analyze financial data from the S&P 500 stocks in the 19-year period 1992–2010. We propose a definition of state for a financial market and use it to identify points of drastic change in the correlation structure. These points are mapped to occurrences of financial crises. We find that a wide variety of characteristic correlation structure patterns exist in the observation time window, and that these characteristic correlation structure patterns can be classified into several typical “market states”. Using this classification we recognize transitions between different market states. A similarity measure we develop thus affords means of understanding changes in states and of recognizing developments not previously seen.

Dyson's correlation functions and graded symmetry

A new derivation of Dyson’s k‐level correlation functions of the Gaussian unitary ensemble (GUE) is given. The method uses matrices with graded symmetry. The number of integrations needed for the ensemble average becomes independent of the level number N. For arbitrary level number N, the k‐level correlation function is expressed as an integral involving the eigenvalues of a 2k×2k graded matrix. The limit of infinitely many levels N→∞ is calculated by a simple saddle‐point approximation of this integral, avoiding the introduction of Hermite polynomials and oscillator wave functions.

Strain in semiconductor core-shell nanowires

We compute strain distributions in core-shell nanowires of zinc blende structure. We use both continuum elasticity theory and an atomistic model, and consider both finite and infinite wires. The atomistic valence force-field (VFF) model has only few assumptions. But it is less computationally efficient than the finite-element (FE) continuum elasticity model. The generic properties of the strain distributions in core-shell nanowires obtained based on the two models agree well. This agreement indicates that although the calculations based on the VFF model are computationally feasible in many cases, the continuum elasticity theory suffices to describe the strain distributions in large core-shell nanowire structures. We find that the obtained strain distributions for infinite wires are excellent approximations to the strain distributions in finite wires, except in the regions close to the ends. Thus, our most computationally efficient model, the FE continuum elasticity model developed for infinite wires, is s...

An Itzykson–Zuber‐like integral and diffusion for complex ordinary and supermatrices

We compute an analogue of the Itzykson–Zuber integral for the case of arbitrary complex matrices. The calculation is done for both ordinary and supermatrices by transferring the Itzykson–Zuber diffusion equation method to the space of arbitrary complex matrices. The integral is of interest for applications in quantum chromodynamics and the theory of two‐dimensional quantum gravity.

Isospin mixing and spectral fluctuation properties

Abstract The breaking of isospin symmetry influences the statistical spectral fluctuations considerably, as shown by a recent experimental analysis of the low energy spectrum of 26 Al by Mitchell et al. A random matrix model for the mixing effect is constructed. Numerical and analytical calculations in the framework of this model provide an explanation and description of the experimental results, and a determination of the r.m.s. Coulomb matrix element.

Microscopic spectrum of the QCD Dirac operator with finite quark masses

We compute the microscopic spectrum of the QCD Dirac operator in the presence of dynamical fermions in the framework of random-matrix theory for the chiral Gaussian unitary ensemble. We obtain results for the microscopic spectral correlators, the microscopic spectral density, and the distribution of the smallest eigenvalue for an arbitrary number of flavors, arbitrary quark masses, and arbitrary topological charge.

Universal spectral correlations of the Dirac operator at finite temperature

Abstract Using the graded eigenvalue method a rea recently computed extension of the Itzykysn-Zuber integral to complex matrices, we compute the k-point spectral correlation functions of the Dirac operator in a chiral random matrix model with a deterministic diagonal matrix added. We obtain results both on the scale of the mean level spacing and on the microscopic scale. We find that the microscopic spectral correlations have the same functional form as at zero temperature, provided that the microscopic variables are rescaled by the temperature-dependent chiral condensate.

Impact of the tick-size on financial returns and correlations

We demonstrate that the lowest possible price change (tick-size) has a large impact on the structure of financial return distributions. It induces a microstructure as well as possibly altering the tail behavior. On small return intervals, the tick-size can distort the calculation of correlations. This especially occurs on small return intervals and thus contributes to the decay of the correlation coefficient towards smaller return intervals (Epps effect). We study this behavior within a model and identify the effect in market data. Furthermore, we present a method to compensate this purely statistical error.

Crossover to non-universal microscopic spectral fluctuations in lattice gauge theory

Abstract The spectrum of the Dirac operator near zero virtuality obtained in lattice gauge simulations is known to be universally described by chiral random matrix theory. We address the question of the maximum energy for which this universality persists. For this purpose, we analyze large ensembles of complete spectra of the Euclidean Dirac operator for staggered fermions. We calculate the disconnected scalar susceptibility and the microscopic number variance for the chiral symplectic ensemble of random matrices and compare the results with lattice Dirac spectra for quenched SU(2). The crossover to a non-universal regime is clearly identified and found to scale with the square of the linear lattice size and with f π 2 , in agreement with theoretical expectations.