abstract

We summarise recent results for the chiral Random Two-Matrix Theory constructed to describe QCD in the epsilon-regime with imaginary chemical potential. The virtue of this theory is that unquenched Lattice simulations can be used to determine both low energy constants \({\mit \Sigma }\) and \(F\) in the leading order chiral Lagrangian, due to their respective coupling to quark mass and chemical potential. We briefly recall the analytic formulas for all density and individual eigenvalue correlations and then illustrate them in detail in the simplest, quenched case with imaginary isospin chemical potential. Some peculiarities are pointed out for this example: (i) the factorisation of density and individual eigenvalue correlation functions for large chemical potential and (ii) the factorisation of the non-Gaussian weight function of bi-orthogonal polynomials into Gaussian weights with ordinary orthogonal polynomials.

abstract

We show how non-compact space-time (ZZ branes) emerges as a limit of compact space-time (FZZT branes) for specific ratios between the square of the boundary cosmological constant and the bulk cosmological constant in the \((2,2m - 1)\) minimal model coupled to two-dimensional Euclidean quantum gravity. Furthermore, we show that the principal \((r,s)\) ZZ brane can be viewed as the basic (1,1) ZZ boundary state tensored with a (\(r,s\)) Cardy boundary state for a general \((p,q)\) minimal model coupled to two-dimensional quantum gravity. In this sense there exists only one ZZ boundary state, the basic (1,1) boundary state.

abstract

We study a new ensemble of random correlation matrices related to multivariate Student (or more generally elliptic) random variables. We establish the exact density of states of empirical correlation matrices that generalizes the Marčenko–Pastur result. The comparison between the theoretical density of states in the Student case and empirical financial data is surprisingly good, even if we are still able to detect systematic deviations. Finally, we compute explicitely the Kullback–Leibler entropies of empirical Student matrices, which are found to be independent of the true correlation matrix, as in the Gaussian case. We provide numerically exact values for these Kullback–Leibler entropies.

abstract

In order to pursue the issue of the relation between the financial cross-correlations and the conventional Random Matrix Theory we analyse several characteristics of the stock market correlation matrices like the distribution of eigenvalues, the cross-correlations among signs of the returns, the volatility cross-correlations, and the multifractal characteristics of the principal values. The results indicate that the stock market dynamics is not simply decomposable into ‘market’, ‘sectors’, and the Wishart random bulk. This clearly is seen when the time series used to construct the correlation matrices are sufficiently long and thus the measurement noise suppressed. Instead, a hierarchically convoluted and highly nonlinear organization of the market emerges and indicates that the relevant information about the whole market is encoded already in its constituents.

abstract

The relation between the Calogero model, namely, a system of \(N\) identical particles in one dimension with inverse-square interactions, and the three classical types of quantum-mechanical matrix models is well-known. In this talk I explore various generalized Calogero models and identify the quantum mechanical matrix model they correspond to at special values of their couplings. I also present and briefly discuss the collective field formulation of these generalized Calogero models.

abstract

We discuss recent results of the replica approach to statistical mechanics of a single classical particle placed in a random \(N (\gg 1)\)-dimensional Gaussian landscape. The particular attention is paid to the case of landscapes with logarithmically growing correlations and to its recent generalizations. Those landscapes give rise to a rich multifractal spatial structure of the associated Boltzmann–Gibbs measure. We also briefly mention related results on counting stationary points of random Gaussian surfaces, as well as ongoing research on statistical mechanics in a random landscape constructed locally by adding many squared Gaussian-distributed terms.

abstract

We compute \(\langle \det (Iz-H)(Iz-H)^{\dagger } \rangle _H\) in the limit of infinite matrix dimension \(N\) for complex random matrices \(H\) with invariant matrix distribution in terms of the eigenvalue distribution of the Hermitian random matrices \(HH^{\dagger }\). Under the assumption that \(\frac {1}{N}\ln \langle \det (Iz-H)(Iz-H)^{\dagger } \rangle _H\) is asymptotically equal to \(\frac {1}{N} \langle \ln \det (Iz-H)(Iz-H)^{\dagger } \rangle _H\) we reproduce the eigenvalue distribution of \(H\) obtained previously by Feinberg and Zee, Nucl. Phys. B501, 643 (1997).

abstract

The problem of filtering information from large correlation matrices is of great importance in many applications. We have recently proposed the use of the Kullback–Leibler distance to measure the performance of filtering algorithms in recovering the underlying correlation matrix when the variables are described by a multivariate Gaussian distribution. Here we use the Kullback–Leibler distance to investigate the performance of filtering methods based on Random Matrix Theory and on the shrinkage technique. We also present some results on the application of the Kullback–Leibler distance to multivariate data which are non Gaussian distributed.

abstract

We review a new technique for calculating spectral properties of infinite non-Hermitian random matrix models, and we present an algorithm for calculating bulk spectral properties of ensembles of the type \(H_{1} + i H_{2}\), where \(H_{1}\) and \(H_{2}\) are arbitrary free (in the sense of Voiculescu) ensembles, including cases of the Lévy (heavy-tailed) spectra. As a particular example, we solve analytically the ensemble \(C_{1} + i C_{2}\), where \(C_{1}\) and \(C_{2}\) are free centered random matrix ensembles of the Cauchy class.

abstract

We give a constructive proof for the superbosonization formula for invariant random matrix ensembles, which is the supersymmetry analog of the theory of Wishart matrices. Formulas are given for unitary, orthogonal and symplectic symmetry, but worked out explicitly only for the orthogonal case. The method promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non-Gaussian type.

abstract

We derive the exact form of the eigenvalue spectrum of correlation matrices obtained from a set of \(N\) time-shifted, iid Gaussian time-series of length \(T\). These matrices are random, real and asymmetric matrices with a superimposed structure due to the time-lag. We demonstrate that the associated (complex) eigenvalue spectrum is circular symmetric for large matrices (\(\lim N \to \infty \)). This fact allows to exactly compute the eigenvalue density via the inverse Abel-transform of the density of the symmetrized problem. The validity of the approach is demonstrated by comparison to numerical realizations of random time-series. As an example, spectra of correlation matrices from time-lagged financial data are presented.

abstract

We discuss the sign problem in QCD at nonzero chemical potential and its relation with chiral symmetry breaking and the spectrum of the Dirac operator using the framework of chiral random matrix theory. We show that the Banks–Casher formula is not valid for theories with a sign problem and has to be replaced by an alternative mechanism that is worked out in detail for QCD in one dimension at nonzero chemical potential.

abstract

Large deviations of the maximum eigenvalue to the left of the expected value are investigated for the Gaussian and Wishart random matrices. Universal rate functions can be computed analytically with a Coulomb gas approach and numerical simulations are in good agreement with the theoretical predictions. In contrast with the case of independent random variables, the exponential decay of the probability of extreme events follows a power \(N^2\) and not \(N\) due to the peculiar level repulsion.