Fractional Brownian motion with Hurst index H=0 and the Gaussian Unitary Ensemble

The goal of this paper is to establish a relation between characteristic polynomials of N×N GUE random matrices H as N→∞, and Gaussian processes with logarithmic correlations. We introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of DN(z)=−log|det(H−zI)| on mesoscopic scales as N→∞. By employing a Fourier integral representation, we use this to prove a continuous analogue of a result by Diaconis and Shahshahani [J. Appl. Probab. 31A (1994) 49–62]. On the macroscopic scale, DN(x) gives rise to yet another type of Gaussian process with logarithmic correlations. We give an explicit construction of the latter in terms of a Chebyshev–Fourier random series

Universal K-matrix distribution in beta=2 ensembles of random matrices

11 pages; published version (added proportionality constants, minor changes)YVF and AN were supported by EPSRC grant EP/J002763/1 'Insights into Disordered Landscapes via Random Matrix Theory and Statistical Mechanics'

Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles

We consider a family of chiral non-Hermitian Gaussian random matrices in the unitarily invariant symmetry class. The eigenvalue distribution in this model is expressed in terms of Laguerre polynomials in the complex plane. These are orthogonal with respect to a non-Gaussian weight including a modified Bessel function of the second kind, and we give an elementary proof for this. In the large n limit, the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between Airy and Poisson that was recently found by one of the authors for the elliptic Ginibre ensemble. We conclude that this scaling limit is universal, appearing for two different non-Hermitian random matrix ensembles with unitary symmetry. As a second result we give an equivalent form for the interpolating Airy kernel in terms of a single real integral, similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum. This makes its structure as a one-parameter deformation of the Airy kernel more transparent

Almost-Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre eigenvalue statistics

By using the method of orthogonal polynomials we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from Hermitian matrices characterized by the Wigner- Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre. Two-point statistical measures (as e.g. spectral form factor, number variance and small distance behavior of the nearest neighbor distance distribution $p(s)$) are studied in more detail. In particular, we found that the latter function may exhibit unusual behavior $p(s)\propto s^{5/2}$ for some parameter values.Comment: 4 pages, RevTE

A few remarks on colour-flavour transformations, truncations of random unitary matrices, Berezin reproducing kernels and Selberg-type integrals

The Humboldt Foundation is acknowledged for the financial support of that visit. The research in Nottingham was supported by EPSRC grant EP/C515056/1 'Random Matrices and Polynomials: a tool to understand complexity'

One-component plasma on a spherical annulus and a random matrix ensemble

The two-dimensional one-component plasma at the special coupling \beta = 2 is known to be exactly solvable, for its free energy and all of its correlations, on a variety of surfaces and with various boundary conditions. Here we study this system confined to a spherical annulus with soft wall boundary conditions, paying special attention to the resulting asymptotic forms from the viewpoint of expected general properties of the two-dimensional plasma. Our study is motivated by the realization of the Boltzmann factor for the plasma system with \beta = 2, after stereographic projection from the sphere to the complex plane, by a certain random matrix ensemble constructed out of complex Gaussian and Haar distributed unitary matrices.Comment: v2, typos and references corrected, 24 pages, 1 figur

Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble we recover the SOP of Forrester and Nagao in terms of Hermite polynomials

Schur function averages for the real Ginibre ensemble

We derive an explicit simple formula for expectations of all Schur functions in the real Ginibre ensemble. It is a positive integer for all entries of the partition even and zero otherwise. The result can be used to determine the average of any analytic series of elementary symmetric functions by Schur function expansion

Eigenvectors under a generic perturbation: non-perturbative results from the random matrix approach

We consider eigenvectors of the Hamiltonian H0 perturbed by a generic perturbation V modelled by a random matrix from the Gaussian Unitary Ensemble (GUE). Using the super-symmetry approach we derive analytical results for the statistics of the eigenvectors, which are non-perturbative in V and valid for an arbitrary deterministic H0. Further we generalise them to the case of a random H0, focusing, in particular, on the Rosenzweig-Porter model. Our analytical predictions are confirmed by numerical simulations

Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature

We consider N non-interacting fermions in an isotropic d-dimensional harmonic trap. We compute analytically the cumulative distribution of the maximal radial distance of the fermions from the trap center at zero temperature. While in d = 1 the limiting distribution (in the large N limit), properly centered and scaled, converges to the squared Tracy–Widom distribution of the Gaussian unitary ensemble in random matrix theory, we show that for all d > 1, the limiting distribution converges to the Gumbel law.These limiting forms turn out to be universal, i.e. independent of the details of the trapping potential for a large class of isotropic trapping potentials. We also study the position of the right-most fermion in a given direction in d dimensions and, in the case of a harmonic trap, the maximum momentum, and show that they obey similar Gumbel statistics. Finally, we generalize these results to low but finite temperature