We extend the so-called "single ring theorem", also known as the Haagerup-Larsen theorem, by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian operator $X$, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of $X$. We show that this object allows to calculate the conditional expectation of the squared eigenvalue condition number. We give examples and we provide cross-check of the analytic prediction by the large scale numerics.

J. Phys. A: Math. Theor. 50 (2017) 105204

DOI: 10.1088/1751-8121/aa5451

Maximally Entangled Multipartite States: A Brief Survey

Marco Enríquez, Iwona Wintrowicz, Karol Życzkowski

The problem of identifying maximally entangled quantum states of a composite quantum systems is analyzed. We review some states of multipartite systems distinguished with respect to certain measures of quantum entanglement. Numerical results obtained for 4-qubit pure states illustrate the fact that the notion of maximally entangled state depends on the measure used.

J. Phys.: Conf. Ser. 698 (2016) 012003

DOI: 10.1088/1742-6596/698/1/012003

Quantum noise generated by local random Hamiltonians

Marcin Markiewicz, Zbigniew Puchała, Anna de Rosier, Wiesław Laskowski, Karol Życzkowski

We investigate the impact of a local random unitary noise on multipartite quantum states of arbitrary dimension. We follow the dynamical approach, in which the single-particle unitaries are generated by local random Hamiltonians. Assuming short evolution time we derive an upper bound on the fidelity between an initial and the final state transformed by this type of noise. This result is based on averaging the Tamm-Mandelstam bound and holds for a wide class of distributions of random Hamiltonians fulfilling specific symmetry conditions. It is showed that the sensitivity of a given pure quantum state to the discussed type of noise depends only on the properties of a single-particle and bipartite reduced states.

Spectra of large time-lagged correlation matrices from Random Matrix
Theory

We analyze the spectral properties of large, time-lagged correlation matrices using the tools of random matrix theory. We compare predictions of the one-dimensional spectra, based on approaches already proposed in the literature. Employing the methods of free random variables and diagrammatic techniques, we solve a general random matrix problem, namely the spectrum of a matrix $\frac{1}{T}XAX^{\dagger}$, where $X$ is an $N\times T$ Gaussian random matrix and $A$ is \textit{any} $T\times T$, not necessarily symmetric (Hermitian) matrix. As a particular application, we present the spectral features of the large lagged correlation matrices as a function of the depth of the time-lag. We also analyze the properties of left and right eigenvector correlations for the time-lagged matrices. We positively verify our results by the numerical simulations.

We give qualitative arguments for the mesoscopic nature of the Sachdev-Yee-Kitaev (SYK) model in the holographic regime with $q^2/N\ll 1$ with $N$ Majorana particles coupled by antisymmetric and random interactions of range $q$. Using a stochastic deformation of the SYK model, we show that its characteristic determinant obeys a viscid Burgers equation with a small spectral viscosity in the opposite regime with $q/N=1/2$, in leading order. The stochastic evolution of the SYK model can be mapped onto that of random matrix theory, with universal Airy oscillations at the edges. A spectral hydrodynamical estimate for the relaxation of the collective modes is made.

Almost all quantum channels are equidistant

Ion Nechita, Zbigniew Puchała, Łukasz Pawela, Karol Życzkowski

In this work we analyze properties of generic quantum channels in the case of large system size. We use the random matrix theory and free probability to show that the distance between two independent random channels tends to a constant value as the dimension of the system grows larger. As a measure of the distance we use the diamond norm. In the case of a flat Hilbert-Schmidt distribution on quantum channels, we obtain that the distance converges to $\frac12 + \frac{2}{\pi}$. Furthermore, we show that for a random state $\rho$ acting on a bipartite Hilbert space $\HH_A \otimes \HH_B$, sampled from the Hilbert-Schmidt distribution, the reduced states $\Tr_A\rho$ and $\Tr_B \rho$ are arbitrarily close to the maximally mixed state. This implies that, for large dimensions, the state $\rho$ may be interpreted as a Jamio{\l}kowski state of a unital map.

Convex set of quantum states with positive partial transpose analysed by
hit and run algorithm

Konrad Szymański, Benoît Collins, Tomasz Szarek, Karol Życzkowski

The convex set of quantum states of a composite $K \times K$ system with positive partial transpose is analysed. A version of the hit and run algorithm is used to generate a sequence of random points covering this set uniformly and an estimation for the convergence speed of the algorithm is derived. For $K\ge 3$ this algorithm works faster than sampling over the entire set of states and verifying whether the partial transpose is positive. The level density of the PPT states is shown to differ from the Marchenko-Pastur distribution, supported in [0,4] and corresponding asymptotically to the entire set of quantum states. Based on the shifted semi--circle law, describing asymptotic level density of partially transposed states, and on the level density for the Gaussian unitary ensemble with constraints for the spectrum we find an explicit form of the probability distribution supported in [0,3], which describes well the level density obtained numerically for PPT states.

Chiral Disorder and Random Matrix Theory with Magnetism

Maciej A. Nowak, Mariusz Sadzikowski, Ismail Zahed

We revisit the concept of chiral disorder in QCD in the presence of a QED magnetic field |eH|. Weak magnetism corresponds to $|eH|≤ 1∕ρ2$ with $ρ ≈ 1∕3$ fm the vacuum instanton size, while strong magnetism the reverse. Asymptotics (ultra-strong magnetism) is in the realm of perturbative QCD. We analyze weak magnetism using the concept of the quark return probability in the diffusive regime of chiral disorder. The result is in agreement with expectations from chiral perturbation theory. We analyze strong and ultra-strong magnetism in the ergodic regime using random matrix theory including the effects of finite temperature. The strong magnetism results are in agreement with the currently reported lattice data in the presence of a small shift of the Polyakov line. The ultra-strong magnetism results are consistent with expectations from the perturbative QCD. We suggest a chiral random matrix effective action with matter and magnetism to analyze the QCD phase diagram near the critical points under the influence of magnetism.

Acta Phys. Pol. B 47 (2016) 2173

DOI: 10.5506/aphyspolb.47.2173

Chiral random matrix model at finite chemical potential: Characteristic determinant and edge universality

We derive an exact formula for the stochastic evolution of the characteristic determinant of a class of deformed Wishart matrices following from a chiral random matrix model of QCD at finite chemical potential. In the WKB approximation, the characteristic determinant describes a sharp droplet of eigenvalues that deforms and expands at large stochastic times. Beyond the WKB limit, the edges of the droplet are fuzzy and described by universal edge functions. At the chiral point, the characteristic determinant in the microscopic limit is universal. Remarkably, the physical chiral condensate at finite chemical potential may be extracted from current and quenched lattice Dirac spectra using the universal edge scaling laws, without having to solve the QCD sign problem.

Nuclear Physics B 909 (2016) 14-42

DOI: 10.1016/j.nuclphysb.2016.04.040

Seeking a fingerprint: analysis of point processes in actigraphy recording

Ewa Gudowska-Nowak, Jeremi K. Ochab, Katarzyna Oleś, Ewa Beldzik, Dante R. Chialvo, Aleksandra Domagalik, Magdalena Fąfrowicz, Tadeusz Marek, Maciej A. Nowak, Halszka Ogińska, Jerzy Szwed, Jacek Tyburczyk

Motor activity of humans displays complex temporal fluctuations which can be characterised by scale-invariant statistics, thus demonstrating that structure and fluctuations of such kinetics remain similar over a broad range of time scales. Previous studies on humans regularly deprived of sleep or suffering from sleep disorders predicted a change in the invariant scale parameters with respect to those for healthy subjects. In this study we investigate the signal patterns from actigraphy recordings by means of characteristic measures of fractional point processes. We analyse spontaneous locomotor activity of healthy individuals recorded during a week of regular sleep and a week of chronic partial sleep deprivation. Behavioural symptoms of lack of sleep can be evaluated by analysing statistics of duration times during active and resting states, and alteration of behavioural organisation can be assessed by analysis of power laws detected in the event count distribution, distribution of waiting times between consecutive movements and detrended fluctuation analysis of recorded time series. We claim that among different measures characterising complexity of the actigraphy recordings and their variations implied by chronic sleep distress, the exponents characterising slopes of survival functions in resting states are the most effective biomarkers distinguishing between healthy and sleep-deprived groups.

J. Stat. Mech. 2016 (2016) 054034

DOI: 10.1088/1742-5468/2016/05/054034

Ornstein–Uhlenbeck diffusion of hermitian and non-hermitian matrices—unexpected links

Jean-Paul Blaizot, Jacek Grela, Maciej A. Nowak, Wojciech Tarnowski, Piotr Warchoł

We compare the Ornstein–Uhlenbeck process for the Gaussian unitary ensemble to its non-hermitian counterpart—for the complex Ginibre ensemble. We exploit the mathematical framework based on the generalized Green's functions, which involves a new, hidden complex variable, in comparison to the standard treatment of the resolvents. This new variable turns out to be crucial to understand the pattern of the evolution of non-hermitian systems. The new feature is the emergence of the coupling between the flow of eigenvalues and that of left/right eigenvectors. We analyze local and global equilibria for both systems. Finally, we highlight some unexpected links between both ensembles.

J. Stat. Mech. 2016 (2016) 054037

DOI: 10.1088/1742-5468/2016/05/054037

Asymptotic entropic uncertainty relations

Radosław Adamczak, Rafał Latała, Zbigniew Puchała, Karol Życzkowski

We analyze entropic uncertainty relations for two orthogonal measurements on a $N$-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix $U$ relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than these obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to establish the conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of an independent interest.

Journal of Mathematical Physics 57 (2016) 032204

DOI: 10.1063/1.4944425

Multipartite entanglement in heterogeneous systems

Dardo Goyeneche, Jakub Bielawski, Karol Życzkowski

Heterogeneous bipartite quantum pure states, composed of two subsystems with a different number of levels, cannot have both reductions maximally mixed. In this work, we demonstrate existence of a wide range of highly entangled states of heterogeneous multipartite systems consisting of $N>2$ parties such that every reduction to one and two parties is maximally mixed. Two constructions of generating genuinely multipartite maximally entangled states of heterogeneous systems for an arbitrary number of subsystems are presented. Such states are related to quantum error correction codes over mixed alphabets and mixed orthogonal arrays. Additionally, we show the advantages of considering heterogeneous systems in practical implementations of multipartite steering.

Phys. Rev. A 94 (2016) 012346

DOI: 10.1103/PhysRevA.94.012346

Hydrodynamics of the Polyakov line in SU(Nc) Yang–Mills

We discuss a hydrodynamical description of the eigenvalues of the Polyakov line at large but finite $N_c$ for Yang-Mills theory in even and odd space-time dimensions. The hydro-static solutions for the eigenvalue densities are shown to interpolate between a uniform distribution in the confined phase and a localized distribution in the de-confined phase. The resulting critical temperatures are in overall agreement with those measured on the lattice over a broad range of $N_c$, and are consistent with the string model results at $N_c=\infty$. The stochastic relaxation of the eigenvalues of the Polyakov line out of equilibrium is captured by a hydrodynamical instanton. An estimate of the probability of formation of a Z(N$_c)$ bubble using a piece-wise sound wave is suggested.

We derive a hydrodynamical description of the eigenvalues of the chiral Dirac spectrum in the vacuum and in the large $N$ (volume) limit. The linearized hydrodynamics supports sound waves. The stochastic relaxation of the eigenvalues is captured by a hydrodynamical instanton configuration which follows from a pertinent form of Euler equation. The relaxation from a phase of localized eigenvalues and unbroken chiral symmetry to a phase of de-localized eigenvalues and broken chiral symmetry occurs over a time set by the speed of sound. We show that the time is $\Delta \tau=\pi\rho(0)/2\beta N$ with $\rho(0)$ the spectral density at zero virtuality and $\beta=1,2,4$ for the three Dyson ensembles that characterize QCD with different quark representations in the ergodic regime.

Phys. Lett. B 753 (2016) 303

DOI: 10.1016/j.physletb.2015.12.032

Hydrodynamical spectral evolution for random matrices

The eigenvalues of the matrix structure $X + X^{(0)}$, where $X$ is a random Gaussian Hermitian matrix and $X^{(0)}$ is non-random or random independent of $X$, are closely related to Dyson Brownian motion. Previous works have shown how an infinite hierarchy of equations satisfied by the dynamical correlations become triangular in the infinite density limit, and give rise to the complex Burgers equation for the Green's function of the corresponding one-point density function. We show how this and analogous partial differential equations, for chiral, circular and Jacobi versions of Dyson Brownian motion follow from a macroscopic hydrodynamical description involving the current density and continuity equation. The method of characteristics gives a systematic approach to solving the PDEs, and in the chiral case we show how this efficiently reclaims the characterisation of the global eigenvalue density for non-central Wishart matrices due to Dozier and Silverstein. Collective variables provide another approach to deriving the complex Burgers equation in the Gaussian case, and we show that this approach applies equally as well to chiral matrices. We relate both the Gaussian and chiral cases to the asymptotics of matrix integrals.

J. Phys. A: Math. Theor. 49 (2016) 085203

DOI: 10.1088/1751-8113/49/8/085203

Hydrodynamical description of the QCD Dirac spectrum at finite chemical potential

We present a hydrodynamical description of the QCD Dirac spectrum at finite chemical potential as an uncompressible droplet in the complex eigenvalue space. For a large droplet, the fluctuation spectrum around the hydrostatic solution is gapped by a longitudinal Coulomb plasmon, and exhibits a frictionless odd viscosity. The stochastic relaxation time for the restoration/breaking of chiral symmetry is set by twice the plasmon frequency. The leading droplet size correction to the relaxation time is fixed by a universal odd viscosity to density ratio $\eta_O/\rho_0=(\beta-2)/4$ for the three Dyson ensembles $\beta=1,2,4$.

We introduce a simple yet powerful calculational tool useful in calculating averages of ratios and products of characteristic polynomials. The method is based on Dyson Brownian motion and Grassmann integration formula for determinants. It is intended as an alternative to other RMT techniques applicable to general gaussian measures. Resulting formulas are exact for finite matrix size N and form integral representations convenient for large N asymptotics. Quantities obtained by the method can be interpreted as averages over matrix models with an external source. We provide several explicit and novel calculations showing a range of applications.

J. Phys. A: Math. Theor. 49 (2015) 015201

DOI: 10.1088/1751-8113/49/1/015201

How we move is universal: Scaling in the average shape of human activity

Dante R. Chialvo, Ana Maria Gonzalez Torrado, Ewa Gudowska-Nowak, Jeremi K. Ochab, Pedro Montoya, Maciej A. Nowak, Enzo Tagliazucchi

Human motor activity is constrained by the rhythmicity of the 24 hours circadian cycle, including the usual 12-15 hours sleep-wake cycle. However, activity fluctuations also appear over a wide range of temporal scales, from days to a few seconds, resulting from the concatenation of a myriad of individual smaller motor events. Furthermore, individuals present different propensity to wakefulness and thus to motor activity throughout the circadian cycle. Are activity fluctuations across temporal scales intrinsically different, or is there a universal description encompassing them? Is this description also universal across individuals, considering the aforementioned variability? Here we establish the presence of universality in motor activity fluctuations based on the empirical study of a month of continuous wristwatch accelerometer recordings. We study the scaling of average fluctuations across temporal scales and determine a universal law characterized by critical exponents $\alpha$, $\tau$ and $1/{\mu}$. Results are highly reminiscent of the universality described for the average shape of avalanches in systems exhibiting crackling noise. Beyond its theoretical relevance, the present results can be important for developing objective markers of healthy as well as pathological human motor behavior.

Papers in Physics, vol. 7, art. 070017 (2015)

DOI: 10.4279/PIP.070017

Quaternionic R transform and non-Hermitian random matrices

Using the Cayley-Dickson construction we rephrase and review the non-hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp and I. Zahed, Nucl.Phys. B $\textbf{501}$, 603 (1997)], that generalizes the free probability calculus to asymptotically large non-hermitian random matrices. The main object in this generalization is a quaternionic extension of the R transform which is a generating function for planar (non-crossing) cumulants. We demonstrate that the quaternionic R transform generates all connected averages of all distinct powers of $X$ and its hermitian conjugate $X^\dagger$: $\langle\langle \frac{1}{N} \mbox{Tr} X^{a} X^{\dagger b} X^c \ldots \rangle\rangle$ for $N\rightarrow \infty$. We show that the R transform for gaussian elliptic laws is given by a simple linear quaternionic map $\mathcal{R}(z+wj) = x + \sigma^2 \left(\mu e^{2i\phi} z + w j\right)$ where $(z,w)$ is the Cayley-Dickson pair of complex numbers forming a quaternion $q=(z,w)\equiv z+ wj$. This map has five real parameters $\Re e x$, $\Im m x$, $\phi$, $\sigma$ and $\mu$. We use the R transform to calculate the limiting eigenvalue densities of several products of gaussian random matrices.

Phys. Rev. E 92, 052111 (2015)

DOI: 10.1103/PhysRevE.92.052111

Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups

The main purpose of this paper of the paper is an explicite construction of generalized Gaussian process with function $t_b(V)=b^{H(V)}$, where $H(V)=n-h(V)$, $h(V)$ is the number of singletons in a pair-partition $V \in P_2(2n)$. This gives another proof of Theorem of A. Buchholtz \cite{Buch} that $t_b$ is positive definite function on the set of all pair-partitions. Some new combinatorial formulas are also presented. Connections with free additive convolutions probability measure on $\mathbb{R}$ are also done. Also new positive definite functions on permutations are presented and also it is proved that the function $H$ is norm (on the group $S(\infty)=\bigcup S(n)$.

We introduce the notion of time reversal in open quantum systems as represented by linear quantum operations, and a related generalization of classical entropy production in the environment. This functional is the ratio of the probability to observe a transition between two states under the forward and the time reversed dynamics, and leads, as in the classical case, to fluctuation relations as tautological identities. As in classical dynamics in contact with a heat bath, time reversal is not unique, and we discuss several possibilities. For any bistochastic map its dual map preserves the trace and describes a legitimate dynamics reversed in time, in that case the entropy production in the environment vanishes. For a generic stochastic map we construct a simple quantum operation which can be interpreted as a time reversal. For instance, the decaying channel, which sends the excited state into the ground state with a certain probability, can be reversed into the channel transforming the ground state into the excited state with the same probability.

J. Phys. A 48, 38FT01 (2015)

DOI: 10.1088/1751-8113/48/38/38FT01

Comparison of eigeninference based on one- and two-point Green's functions

Zbigniew Drogosz, Jerzy Jurkiewicz, Grzegorz Łukaszewski, Maciej A. Nowak

We compare two methods of eigen-inference from large sets of data, based on the analysis of one-point and two-point Green's functions, respectively. Our analysis points at the superiority of eigen-inference based on one-point Green's function. First, the applied by us method based on Pad?e approximants is orders of magnitude faster comparing to the eigen-inference based on uctuations (two-point Green's functions). Second, we have identi?ed the source of potential instability of the two-point Green's function method, as arising from the spurious zero and negative modes of the estimator for a variance operator of the certain multidimensional Gaussian distribution, inherent for the two-point Green's function eigen-inference method. Third, we have presented the cases of eigen-inference based on negative spectral moments, for strictly positive spectra. Finally, we have compared the cases of eigen-inference of real-valued and complex-valued correlated Wishart distributions, reinforcing our conclusions on an advantage of the one-point Green's function method.

Phys. Rev. E 92, 022111 (2015)

DOI: 10.1103/PhysRevE.92.022111

Unveiling the significance of eigenvectors in diffusing non-Hermitian matrices by identifying the underlying Burgers dynamics

Zdzisław Burda, Jacek Grela, Maciej A. Nowak, Wojciech Tarnowski, Piotr Warchoł

Following our recent letter, we study in detail an entry-wise diffusion of non-hermitian complex matrices. We obtain an exact partial differential equation (valid for any matrix size $N$ and arbitrary initial conditions) for evolution of the averaged extended characteristic polynomial. The logarithm of this polynomial has an interpretation of a potential which generates a Burgers dynamics in quaternionic space. The dynamics of the ensemble in the large $N$ is completely determined by the coevolution of the spectral density and a certain eigenvector correlation function. This coevolution is best visible in an electrostatic potential of a quaternionic argument built of two complex variables, the first of which governs standard spectral properties while the second unravels the hidden dynamics of eigenvector correlation function. We obtain general large $N$ formulas for both spectral density and 1-point eigenvector correlation function valid for any initial conditions. We exemplify our studies by solving three examples, and we verify the analytic form of our solutions with numerical simulations.

Nucl. Phys. B 897, 421 (2015)

DOI: 10.1016/j.nuclphysb.2015.06.002

Minimal Rényi–Ingarden–Urbanik Entropy of Multipartite Quantum States

Marco Enríquez, Zbigniew Puchała, Karol Życzkowski

We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with $d$ levels each. It can be described by the R\'enyi-Ingarden-Urbanik entropy $S_q$ of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case $q=0$ this quantity becomes a function of the rank of the tensor representing the state, while in the limit $q \to \infty$ the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system the entropy $S_1$ coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three and four-qubit systems. In the former case the distributions of $3$-tangle is studied and some of its moments are evaluated, while in the latter case we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit.

Properties of random mixed states of order $N$ distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large $N$, due to the concentration of measure, the trace distance between two random states tends to a fixed number ${\tilde D}=1/4+1/\pi$, which yields the Helstrom bound on their distinguishability. To arrive at this result we apply free random calculus and derive the symmetrized Marchenko--Pastur distribution, which is shown to describe numerical data for the model of two coupled quantum kicked tops. Asymptotic values for the fidelity, Bures and transmission distances between two random states are obtained. Analogous results for quantum relative entropy and Chernoff quantity provide other bounds on the distinguishablity of both states in a multiple measurement setup due to the quantum Sanov theorem.

Phys. Rev. A 93 (2016) 062112

DOI: 10.1103/physreva.93.062112

Spectral density of generalized Wishart matrices and free multiplicative convolution

Wojciech Młotkowski, Maciej A. Nowak, Karol A. Penson, Karol Życzkowski

We investigate the level density for several ensembles of positive random matrices of a Wishart--like structure, $W=XX^{\dagger}$, where $X$ stands for a nonhermitian random matrix. In particular, making use of the Cauchy transform, we study free multiplicative powers of the Marchenko-Pastur (MP) distribution, ${\rm MP}^{\boxtimes s}$, which for an integer $s$ yield Fuss-Catalan distributions corresponding to a product of $s$ independent square random matrices, $X=X_1\cdots X_s$. New formulae for the level densities are derived for $s=3$ and $s=1/3$. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.

Phys. Rev. E 92, 012121 (2015)

DOI: 10.1103/PhysRevE.92.012121

Invariant sums of random matrices and the onset of level repulsion

We compute analytically the joint probability density of eigenvalues and the level spacing statistics for an ensemble of random matrices with interesting features. It is invariant under the standard symmetry groups (orthogonal and unitary) and yet the interaction between eigenvalues is not Vandermondian. The ensemble contains real symmetric or complex hermitian matrices $\mathbf{S}$ of the form $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{O}_i \mathbf{D}_i\mathbf{O}_i^{\mathrm{T}}\rangle$ or $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{U}_i \mathbf{D}_i\mathbf{U}_i^\dagger\rangle$ respectively. The diagonal matrices $\mathbf{D}_i=\mathrm{diag}\{\lambda_1^{(i)},\ldots,\lambda_N^{(i)}\}$ are constructed from real eigenvalues drawn \emph{independently} from distributions $p^{(i)}(x)$, while the matrices $\mathbf{O}_i$ and $\mathbf{U}_i$ are all orthogonal or unitary. The average $\langle\cdot\rangle$ is simultaneously performed over the symmetry group and the joint distribution of $\{\lambda_j^{(i)}\}$. We focus on the limits i.) $N\to\infty$ and ii.) $M\to\infty$, with $N=2$. In the limit i.), the resulting sum $\mathbf{S}$ develops level repulsion even though the original matrices do not feature it, and classical RMT universality is restored asymptotically. In the limit ii.) the spacing distribution attains scaling forms that are computed exactly: for the orthogonal case, we recover the $\beta=1$ Wigner's surmise, while for the unitary case an entirely new universal distribution is obtained. Our results allow to probe analytically the microscopic statistics of the sum of random matrices that become asymptotically free. We also give an interpretation of this model in terms of radial random walks in a matrix space. The analytical results are corroborated by numerical simulations.

J. Stat. Mech. 2015 P06024

DOI: 10.1088/1742-5468/2015/06/P06024

Diffusion in the Space of Complex Hermitian Matrices --- Microscopic Properties of the Averaged Characteristic Polynomial and the Averaged Inverse Characteristic Polynomial

Jean-Paul Blaizot, Jacek Grela, Maciej A. Nowak, Piotr Warchoł

We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial differential, diffusion-like, equations. These equations are valid for matrices of arbitrary size. Their solutions can be given an integral representation that allows for a simple study of their asymptotic behaviors for a broad range of initial conditions.

Acta Phys. Pol. B 46(9), 1801 (2015)

DOI: 10.5506/APhysPolB.46.1801

Universal Spectral Shocks in Random Matrix Theory --- Lessons for QCD

Jean-Paul Blaizot, Jacek Grela, Maciej A. Nowak, Piotr Warchoł

Following Dyson, we treat the eigenvalues of a random matrix as a system
of particles undergoing random walks. The dynamics of large matrices
is then well-described by fluid dynamical equations. In particular, the inviscid
Burgers’ equation is ubiquitous and controls the behavior of the spectral
density of large matrices. The solutions to this equation exhibit shocks that
we interpret as the edges of the spectrum of eigenvalues. Going beyond the
large $N$ limit, we show that the average characteristic polynomial (or the
average of the inverse characteristic polynomial) obeys equations that are
equivalent to a viscid Burgers’ equation, or equivalently a diffusion equation,
with $1/N$ playing the role of the viscosity and encoding the entire
finite $N$ effects. This approach allows us to recover in an elementary way
many results concerning the universal behavior of random matrix theories
and to look at QCD spectral features from a new perspective.

Acta Phys. Pol. B 46 (2015) 1785

DOI: 10.5506/aphyspolb.46.1785

Proceedings of "Random Matrix Theory: Foundations and Applications, Conference, Kraków, Poland, July 1-6, 2014"

Maciej A. Nowak, Jeremi K. Ochab

Acta Physica Polonica B46 (5) 1603-1906 (2015)

Teoria macierzy przypadkowych jako rachunek prawdopodobieństwa

Maciej A. Nowak

Rozdział w monografii "Metody matematyczne w zastosowaniach, tom II", wydanym przez Centrum Zastosowań Matematyki (w druku)

Numerical range for random matrices

Benoît Collins, Piotr Gawron, Alexander E. Litvak, Karol Życzkowski

We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. For a large class of random matrices their numerical range is shown to converge to a disc. In particular, numerical range of complex Ginibre matrix almost surely converges to the disk of radius $\sqrt{2}$. Since the spectrum of non-hermitian random matrices from the Ginibre ensemble lives asymptotically in a neighborhood of the unit disk, it follows that the outer belt of width $\sqrt{2}-1$ containing no eigenvalues can be seen as a quantification the non-normality of the complex Ginibre random matrix. We also show that the numerical range of upper triangular Gaussian matrices converges to the same disk of radius $\sqrt{2}$, while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to $\sqrt{2e}$.

J. Math. Anal. Appl. 418 (2014) 516-533

DOI: 10.1016/j.jmaa.2014.03.072

Universal distribution of Lyapunov exponents for products of Ginibre matrices

Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random $N\times N$ matrices also called Ginibre ensemble we rederive the Lyapunov exponents for an infinite product. We show that for a large number $t$ of product matrices the distribution of each Lyapunov exponent is normal and compute its $t$-dependent variance as well as corrections in a $1/t$ expansion. Originally Lyapunov exponents are defined for singular values of the product matrix that represents a linear time evolution. Surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order. We discuss a general mechanism for $2\times 2$ matrices why the singular values and the radii of complex eigenvalues collapse onto the same value in the large-$t$ limit. Thereby we rederive Newman's triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits $t\to\infty$ and $N\to\infty$ on the global and the local scale. As a mathematical byproduct we show that a particular asymptotic expansion of a Meijer G-function with large index leads to a Gaussian.

J. Phys. A: Math. Theor. 47 395202 (2014)

DOI: 10.1088/1751-8113/47/39/395202

Scale-Free Fluctuations in Behavioral Performance: Delineating Changes in Spontaneous Behavior of Humans with Induced Sleep Deficiency

Jeremi K. Ochab, Jacek Tyburczyk, Ewa Beldzik, Dante R. Chialvo, Aleksandra Domagalik, Magdalena Fąfrowicz, Ewa Gudowska-Nowak, Tadeusz Marek, Maciej A. Nowak, Halszka Ogińska, Jerzy Szwed

The timing and dynamics of many diverse behaviors of mammals, e.g., patterns of animal foraging or human communication in social networks exhibit complex self-similar properties reproducible over multiple time scales. In this paper, we analyze spontaneous locomotor activity of healthy individuals recorded in two different conditions: during a week of regular sleep and a week of chronic partial sleep deprivation. After separating activity from rest with a pre-defined activity threshold, we have detected distinct statistical features of duration times of these two states. The cumulative distributions of activity periods follow a stretched exponential shape, and remain similar for both control and sleep deprived individuals. In contrast, rest periods, which follow power-law statistics over two orders of magnitude, have significantly distinct distributions for these two groups and the difference emerges already after the first night of shortened sleep. We have found steeper distributions for sleep deprived individuals, which indicates fewer long rest periods and more turbulent behavior. This separation of power-law exponents is the main result of our investigations, and might constitute an objective measure demonstrating the severity of sleep deprivation and the effects of sleep disorders.

PLoS ONE 9(9): e107542

DOI: 10.1371/journal.pone.0107542

Dysonian Dynamics of the Ginibre Ensemble

Zdzisław Burda, Jacek Grela, Maciej A. Nowak, Wojciech Tarnowski, Piotr Warchoł

We study the time evolution of Ginibre matrices whose elements undergo Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the dynamics of eigenvalues to the evolution of eigenvectors in a non-trivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and we unravel a hidden dynamics as a function of new complex variable, which in the standard description is treated as a regulator only. We solve the evolution equations for large matrices and demonstrate that the non-analytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. We conjecture that the hidden dynamics, that we observe for the Ginibre ensemble, is a general feature of non-Hermitian random matrix models and is relevant to related physical applications.

Phys.Rev.Lett. 113 (2014) 104102

DOI: 10.1103/PhysRevLett.113.104102

Diagonal unitary entangling gates and contradiagonal quantum states

Arul Lakshminarayan, Zbigniew Puchała, Karol Życzkowski

Nonlocal properties of an ensemble of diagonal random unitary matrices of order $N^2$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log N^2$ behavior characteristic to random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $\rho=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square matrix of order $N$. This fact provides a motivation to study the ensemble of non-hermitian unimodular matrices $A$, with all entries of the same modulus and random phases and the ensemble of quantum states $\rho$, such that all their diagonal entries are equal to $1/N$. Such a state is contradiagonal with respect to the computational basis, in sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the "Borel triangle". This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.

Phys. Rev. A 90, 032303 (2014)

DOI: 10.1103/PhysRevA.90.032303

Universal microscopic correlation functions for products of truncated unitary matrices

Gernot Akemann, Zdzisław Burda, Mario Kieburg, Taro Nagao

We investigate the spectral properties of the product of $M$ complex non-Hermitian random matrices that are obtained by removing $L$ rows and columns of larger unitary random matrices uniformly distributed on the group ${\rm U}(N+L)$. Such matrices are called truncated unitary matrices or random contractions. We first derive the joint probability distribution for the eigenvalues of the product matrix for fixed $N,\ L$, and $M$, given by a standard determinantal point process in the complex plane. The weight however is non-standard and can be expressed in terms of the Meijer G-function. The explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large $N$ (and $L$) limits at fixed $M$. At strong non-unitarity, with $L/N$ finite, the eigenvalues condense on a domain inside the unit circle. At the edge and in the bulk we find the same universal microscopic kernel as for a single complex non-Hermitian matrix from the Ginibre ensemble. At the origin we find the same new universality classes labelled by $M$ as for the product of $M$ matrices from the Ginibre ensemble. Keeping a fixed size of truncation, $L$, when $N$ goes to infinity leads to weak non-unitarity, with most eigenvalues on the unit circle as for unitary matrices. Here we find a new microscopic edge kernel that generalizes the known results for M=1. We briefly comment on the case when each product matrix results from a truncation of different size $L_j$.

J. Phys. A: Math. Theor. 47 (2014) 255202

DOI: 10.1088/1751-8113/47/25/255202

Universal shocks in the Wishart random-matrix ensemble. II. Nontrivial initial conditions

We study the diffusion of complex Wishart matrices and derive a partial differential equation governing the behavior of the associated averaged characteristic polynomial. In the limit of large size matrices, the inverse Cole-Hopf transform of this polynomial obeys a nonlinear partial differential equation whose solutions exhibit shocks at the evolving edges of the eigenvalue spectrum. In a particular scenario one of those shocks hits the origin that plays the role of an impassable wall. To investigate the universal behavior in the vicinity of this wall, a critical point, we derive an integral representation for the averaged characteristic polynomial and study its asymptotic behavior. The result is a Bessoid function.

Jarzynski equality and related fluctuation theorems can be formulated for various setups. Such an equality was recently derived for nonunitary quantum evolutions described by unital quantum operations, i.e., for completely positive, trace-preserving maps, which preserve the maximally mixed state. We analyze here a more general case of arbitrary quantum operations on finite systems and derive the corresponding form of the Jarzynski equality. It contains a correction term due to nonunitality of the quantum map. Bounds for the relative size of this correction term are established and they are applied for exemplary systems subjected to quantum channels acting on a finite-dimensional Hilbert space.

Phys. Rev. E 89, 012127 (2014)

DOI: 10.1103/PhysRevE.89.012127

Exponentially convergent algorithm to generate random points in a
$d$--dimensional body

Benoît Collins, Termeh Kousha, Rafał Kulik, Tomasz Szarek, Karol Życzkowski

An algorithm to generate random points inside an arbitrary $d$--dimensional convex body $X$ with respect to the flat (Lebesgue) measure is proposed. It can be considered as an iterated functions system (IFS) with an infinite number of functions acting on $X$. We analyze the corresponding Markov operator which acts on the probability measures, and show that any initial measure converges exponentially to the uniform measure. Estimations for the convergence rate are derived in terms of the dimension $d$ and the ratio between the radius of the sphere inscribed in $X$ and the radius of the outscribed sphere. Concrete estimations are provided for the Birkhoff polytope containing bistochastic matrices, the set of quantum states acting on $N$--dimensional Hilbert space and its subset consisting of states with positive partial transpose.

Free products of large random matrices – a short review of recent developments

We review methods to calculate eigenvalue distributions of products of large random matrices. We discuss a generalization of the law of free multiplication to non-Hermitian matrices and give a couple of examples illustrating how to use these methods in practice. In particular we calculate eigenvalue densities of products of Gaussian Hermitian and non-Hermitian matrices including combinations of GUE and Ginibre matrices.

J. Phys.: Conf. Ser. 473 012002 (2013)

DOI: 10.1088/1742-6596/473/1/012002

An extended anyon Fock space and noncommutative Meixner-type orthogonal
polynomials in infinite dimensions

Let $\nu$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighborhood of zero. An anyon L\'evy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\omega,\varphi\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimes\nu)$. Here $\varphi=\varphi(x)$ runs over a space of test functions on $\mathbb R^d$, while $\omega=\omega(x)$ is interpreted as an operator-valued distribution on $\mathbb R^d$. Let $L^2(\tau)$ be the noncommutative $L^2$-space generated by the algebra of polynomials in variables $\langle \omega,\varphi\rangle$, where $\tau$ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in $L^2(\tau)$ of the form $\langle P_n(\omega),f^{(n)}\rangle$, where $f^{(n)}$ is a test function on $(\mathbb R^d)^n$. Using these orthogonal polynomials, we derive a unitary isomorphism $U$ between $L^2(\tau)$ and an extended anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathbf F(L^2(\mathbb R^d,dx))$. The usual anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathcal F(L^2(\mathbb R^d,dx))$, is a subspace of $\mathbf F(L^2(\mathbb R^d,dx))$. Furthermore, we have the equality $\mathbf F(L^2(\mathbb R^d,dx))=\mathcal F(L^2(\mathbb R^d,dx))$ if and only if the measure $\nu$ is concentrated at one point, i.e., in the Gaussian/Poisson case. Using the unitary isomorphism $U$, we realize the operators $\langle \omega,\varphi\rangle$ as a Jacobi (i.e., tridiagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. We derive a Meixner-type class of anyon L\'evy white noise for which the respective Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure.

Commutative law for products of infinitely large isotropic random matrices

Ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is spectrally commutative and self-averaging in the limit of infinite matrix size $N \rightarrow \infty$. The notion of spectral commutativity means that the eigenvalue density of a product ABC... of such matrices is independent of the order of matrix multiplication, for example the matrix ABCD has the same eigenvalue density as ADCB. In turn, the notion of self-averaging means that the product of n independent but identically distributed random matrices, which we symbolically denote by AAA..., has the same eigenvalue density as the corresponding power A^n of a single matrix drawn from the underlying matrix ensemble. For example, the eigenvalue density of ABCCABC is the same as of A^2B^2C^3. We also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues $\lambda \rightarrow 0$. We show that the singularities at the origin of the eigenvalue density and of the singular value density are in one-to-one correspondence in the limit $N \rightarrow \infty$: the eigenvalue density of an isotropic random matrix has a power law singularity at the origin $\sim |\lambda|^{-s}$ with a power $s \in (0,2)$ when and only when the density of its singular values has a power law singularity $\sim \lambda^{-\sigma}$ with a power $\sigma = s/(4-s)$. These results are obtained analytically in the limit $N \rightarrow \infty$. We supplement these results with numerical simulations for large but finite N and discuss finite size effects for the most common ensembles of isotropic random matrices.

Random pure states of multi-partite quantum systems, associated with arbitrary graphs, are investigated. Each vertex of the graph represents a generic interaction between subsystems, described by a random unitary matrix distributed according to the Haar measure, while each edge of the graph represents a bi-partite, maximally entangled state. For any splitting of the graph into two parts we consider the corresponding partition of the quantum system and compute the average entropy of entanglement. First, in the special case where the partition does not "cross" any vertex of the graph, we show that the area law is satisfied exactly. In the general case, we show that the entropy of entanglement obeys an area law on average, this time with a correction term that depends on the topologies of the graph and of the partition. The results obtained are applied to the problem of distribution of quantum entanglement in a quantum network with prescribed topology.

J. Phys. A: Math. Theor. 46 (2013) 305302

DOI: 10.1088/1751-8113/46/30/305302

Burgers-like equation for spontaneous breakdown of the chiral symmetry in QCD

We link the spontaneous breakdown of chiral symmetry in Euclidean QCD to the collision of spectral shock waves in the vicinity of zero eigenvalue of Dirac operator. The mechanism, originating from complex Burger's-like equation for viscid, pressureless, one-dimensional flow of eigenvalues, is similar to recently observed weak-strong coupling phase transition in large $N_c$ Yang-Mills theory. The spectral viscosity is proportional to the inverse of the size of the random matrix that replaces the Dirac operator in the universal (ergodic) regime. We obtain the exact scaling function and critical exponents of the chiral phase transition for the averaged characteristic polynomial for $N_c \ge3$ QCD. We reinterpret our results in terms of known properties of chiral random matrix models and lattice data.

Physics Letters B, Volume 724, Issue 1-3, p. 170-175

DOI: 10.1016/j.physletb.2013.06.022

Universal shocks in the Wishart random-matrix ensemble

We show that the derivative of the logarithm of the average characteristic polynomial of a diffusing Wishart matrix obeys an exact partial differential equation valid for an arbitrary value of N, the size of the matrix. In the large N limit, this equation generalizes the simple Burgers equation that has been obtained earlier for Hermitian or unitary matrices. The solution through the method of characteristics presents singularities that we relate to the precursors of shock formation in fluid dynamical equations. The 1/N corrections may be viewed as viscous corrections, with the role of the viscosity being played by the inverse of the doubled dimension of the matrix. These corrections are studied through a scaling analysis in the vicinity of the shocks, and one recovers in a simple way the universal Bessel oscillations (so-called hard edge singularities) familiar in random matrix theory.

For delta operator $aD-bD^{p+1}$ we find the corresponding polynomial sequence of binomial type and relations with Fuss numbers. In the case $D-\frac{1}{2}D^2$ we show that the corresponding Bessel-Carlitz polynomials are moments of the convolution semigroup of inverse Gaussian distributions. We also find probability distributions $\nu_{t}$, $t>0$, for which $\left\{y_{n}(t)\right\}$, the Bessel polynomials at $t$, is the moment sequence.

Probability and Mathematical Statistics, (2013) 33.2, 401-408

Gelfand-Raikov representation of Coxeter groups associated with positive definite norm functions

The main purpose of the paper is to study the type of Gelfand–Raikov representations of Coxeter groups $(W, S)$ for the special positive definite functions coming from the deformed Poisson (Haagerup) positive definite functions $q^{L(w)}$ for some special length (norm) functions $L$ on Coxeter groups $W$.

Prob.Math.Stat.34(1),161-180

Better Late than Never: Information Retrieval from Black Holes

Samuel L. Braunstein, Stefano Pirandola, Karol Życzkowski

We show that, in order to preserve the equivalence principle until late times in unitarily evaporating black holes, the thermodynamic entropy of a black hole must be primarily entropy of entanglement across the event horizon. For such black holes, we show that the information entering a black hole becomes encoded in correlations within a tripartite quantum state, the quantum analogue of a one-time pad, and is only decoded into the outgoing radiation very late in the evaporation. This behavior generically describes the unitary evaporation of highly entangled black holes and requires no specially designed evolution. Our work suggests the existence of a matter-field sum rule for any fundamental theory.

Physical Review Letters 110, 101301 (2013)

DOI: 10.1103/PhysRevLett.110.101301

On free infinite divisibility for classical Meixner distributions

We prove that symmetric Meixner distributions, whose probability densities are proportional to $|\Gamma(t+ix)|^2$, are freely infinitely divisible for $0<t\leq\frac{1}{2}$. The case $t=\frac{1}{2}$ corresponds to the law of L\'evy's stochastic area whose probability density is $\frac{1}{\cosh(\pi x)}$. A logistic distribution, whose probability density is proportional to $\frac{1}{\cosh^2(\pi x)}$, is freely infinitely divisible too.

Probab. Math. Stat. 33, Fasc. 2 (2013), 363-375

Collective Correlations of Brodmann Areas fMRI Study with RMT-Denoising

Zdzisław Burda, Jennifer Kornelsen, Maciej A. Nowak, Bartosz Porębski, Uta Sboto-Frankenstein, Bogusław Tomanek, Jacek Tyburczyk

We study collective behavior of Brodmann regions of human cerebral cortex using functional Magnetic Resonance Imaging (fMRI) and Random Matrix Theory (RMT). The raw fMRI data is mapped onto the cortex regions corresponding to the Brodmann areas with the aid of the Talairach coordinates. Principal Component Analysis (PCA) of the Pearson correlation matrix for 41 different Brodmann regions is carried out to determine their collective activity in the idle state and in the active state stimulated by tapping. The collective brain activity is identified through the statistical analysis of the eigenvectors to the largest eigenvalues of the Pearson correlation matrix. The leading eigenvectors have a large participation ratio. This indicates that several Broadmann regions collectively give rise to the brain activity associated with these eigenvectors. We apply random matrix theory to interpret the underlying multivariate data.

We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble (IUE) is equal to the eigenvalue density of n-th power of a single matrix from this ensemble, in the limit when the size of the matrix tends to infinity. Using this observation one can derive the limiting density of the product of n independent identically distributed non-hermitian matrices with unitary invariant measures. In this paper we discuss two examples: the product of n Girko-Ginibre matrices and the product of n truncated unitary matrices. We also provide an evidence that the result holds also for isotropic orthogonal ensembles (IOE).

Phys. Rev. E 86, 061137 (2012)

DOI: 10.1103/PhysRevE.86.061137

Densities of the Raney distributions

Wojciech Młotkowski, Karol A. Penson, Karol Życzkowski

We prove that if $p\ge 1$ and $0< r\le p$ then the sequence $\binom{mp+r}{m}\frac{r}{mp+r}$, $m=0,1,2,...$, is positive definite, more precisely, is the moment sequence of a probability measure $\mu(p,r)$ with compact support contained in $[0,+\infty)$. This family of measures encompasses the multiplicative free powers of the Marchenko-Pastur distribution as well as the Wigner's semicircle distribution centered at $x=2$. We show that if $p>1$ is a rational number, $0<r\le p$, then $\mu(p,r)$ is absolutely continuous and its density $W_{p,r}(x)$ can be expressed in terms of the Meijer and the generalized hypergeometric functions. In some cases, including the multiplicative free square and the multiplicative free square root of the Marchenko-Pastur measure, $W_{p,r}(x)$ turns out to be an elementary function.

Documenta Mathematica vol. 18 (2013), pp. 1593-1596, (24 pages)

Universal microscopic correlation functions for products of independent Ginibre matrices

We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n>1 and generalise the known Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.

J. Phys. A: Math. Theor. 45 (2012) 465201

DOI: 10.1088/1751-8113/45/46/465201

Noncommutative Lévy Processes for Generalized (Particularly Anyon) Statistics

Marek Bożejko, Eugene Lytvynov, Janusz Wysoczański

Let $T=\mathbb R^d$. Let a function $Q:T^2\to\mathbb C$ satisfy $Q(s,t)=\bar{Q(t,s)}$ and $|Q(s,t)|=1$. A generalized statistics is described by creation operators $\partial_t^\dag$ and annihilation operators $\partial_t$, $t\in T$, which satisfy the $Q$-commutation relations. From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which $Q(s,t)$ is equal to $q$ if $s<t$, and to $\bar q$ if $s>t$. Here $q\in\mathbb C$, $|q|=1$. We start the paper with a detailed discussion of a $Q$-Fock space and operators $(\partial_t^\dag,\partial_t)_{t\in T}$ in it, which satisfy the $Q$-commutation relations. Next, we consider a noncommutative stochastic process (white noise) $\omega(t)=\partial_t^\dag+\partial_t+\lambda\partial_t^\dag\partial_t$, $t\in T$. Here $\lambda\in\mathbb R$ is a fixed parameter. The case $\lambda=0$ corresponds to a $Q$-analog of Brownian motion, while $\lambda\ne0$ corresponds to a (centered) $Q$-Poisson process. We study $Q$-Hermite ($Q$-Charlier respectively) polynomials of infinitely many noncommutatative variables $(\omega(t))_{t\in T}$. The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding L\'evy processes. To this end, we recursively define $Q$-cumulants of a field $(\xi(t))_{t\in T}$. This allows us to define a $Q$-L\'evy process as a field $(\xi(t))_{t\in T}$ whose values at different points of $T$ are $Q$-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a $Q$-L\'evy process, and derive a Nualart-Schoutens-type chaotic decomposition for such a process.

Statistics,Comm.Math.Phys. 313(2012),535-569

DOI: 10.1007/s00220-012-1437-8

Deformed Fock spaces, Hecke Operators and Monotone Fock space of Muraki

The main purpose of this paper is to extend our previous construction of T -Fock spaces from a given Yang–Baxter operator satisfying the inequalities −1 ≤ T ≤ 1 to the constructions of T-symmetric Fock spaces related to the class of Yang–Baxter– Hecke operators meeting a weaker condition that T ≥ −1. The new representation of the monotone Fock space of N. Muraki will be given. The main idea of this paper is the new class of generalized Gaussian random variables acting on suitable T-symmetric Fock spaces. Relations with the row and column operator space will be also given.