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We show that the distribution of bulk spacings between pairs of adjacent eigenvalue real parts of a random matrix drawn from the complex elliptic Ginibre ensemble is asymptotically given by a generalization of the Gaudin-Mehta distribution, in the limit of weak non-Hermiticity. The same generalization is expressed in terms of an integro-differential Painlevé function and it is shown that the generalized Gaudin-Mehta distribution describes the crossover, with increasing degree of non-Hermiticity, from Gaudin-Mehta nearest-neighbor bulk statistics in the Gaussian Unitary Ensemble to Poisson gap statistics for eigenvalue real parts in the bulk of the Complex Ginibre Ensemble. [ABSTRACT FROM AUTHOR]

We investigate the joint moments of derivatives of characteristic polynomials over the unitary symplectic group S p (2 N) and the orthogonal ensembles S O (2 N) and O − (2 N) . We prove asymptotic formulae for the joint moments of the n 1th and n 2th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of L -functions with symplectic and orthogonal symmetry. [ABSTRACT FROM AUTHOR]

We consider the asymptotic behaviour of a family of unidimensional lattice fermion models, which are in exact correspondence with certain probability laws on partitions and on unitary matrices. These models exhibit limit shapes, and in the case where the bulk of these shapes are described by analytic functions, the fluctuations around their interfaces have been shown to follow a universal Tracy–Widom distribution or its higher-order analogue. Non-differentiable bulk limit shape functions arise when a gap appears in some quantum numbers of the model, in other words when the Fermi sea is split. We show that split Fermi seas give rise to new interface fluctuations, governed by integer powers of universal distributions. This breakdown in universality is analogous to the behaviour of a random Hermitian matrix when the support of its limiting eigenvalue distribution has multiple cuts, with oscillations appearing in the limit of the two-point correlation function. We show that when the Fermi sea is split in the lattice fermion model, there are multiple cuts in the eigenvalue support of the corresponding unitary matrix model. [ABSTRACT FROM AUTHOR]

We evaluate, in the large- N limit, the complete probability distribution P (A , m) of the values A of the sum ∑ i = 1 N | λ i | m , where λ i ( i = 1 , 2 , ... , N ) are the eigenvalues of a Gaussian random matrix, and m is a positive real number. Combining the Coulomb gas method with numerical simulations using a matrix variant of the Wang–Landau algorithm, we found that, in the limit of N → ∞ , the rate function of P (A , m) exhibits phase transitions of different characters. The phase diagram of the system on the (A, m) plane is surprisingly rich, as it includes three regions: (i) a region with a single-interval support of the optimal spectrum of eigenvalues, (ii) a region emerging for m < 2 where the optimal spectrum splits into two separate intervals, and (iii) a region emerging for m > 2 where the maximum or minimum eigenvalue 'evaporates' from the rest of eigenvalues and dominates the statistics of A. The phase transition between regions (i) and (iii) is of second order. Analytical arguments and numerical simulations strongly suggest that the phase transition between regions (i) and (ii) is of (in general) fractional order p = 1 + 1 / | m − 1 | , where 0 < m < 2 . The transition becomes of infinite order in the special case of m = 1, where we provide a more complete analytical and numerical description. Remarkably, the transition between regions (i) and (ii) for m ⩽ 1 and the transition between regions (i) and (iii) for m > 2 occur at the ground state of the Coulomb gas which corresponds to the Wigner's semicircular distribution. [ABSTRACT FROM AUTHOR]

We study the capacity of entanglement as an alternative to entanglement entropies in estimating the degree of entanglement of quantum bipartite systems over fermionic Gaussian states. In particular, we derive the exact and asymptotic formulas of average capacity of two different cases—with and without particle number constraints. For the later case, the obtained formulas generalize some partial results of average capacity in the literature. The key ingredient in deriving the results is a set of new tools for simplifying finite summations developed very recently in the study of entanglement entropy of fermionic Gaussian states. [ABSTRACT FROM AUTHOR]

Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions converge to a peculiar universal curve p z r s (λ) that looks like a cross between the Wigner semicircle and a Gaussian distribution. An analytic theory for this curve, originally due to Fyodorov, can be developed using supersymmetry-based techniques. We extend these derivations to the case of sparse matrices, including the important case of graph Laplacians for large random graphs with N vertices of mean degree c. In the regime 1 ≪ c ≪ N , the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of p z r s (λ) , centered at c with width ∼ c . At smaller c, this curve receives corrections in powers of 1 / c accurately captured by our theory. For the normalized graph Laplacian (diffusion with a fixed transition rate per vertex), the large c limit is a shifted and scaled Wigner semicircle, again with corrections captured by our analysis. [ABSTRACT FROM AUTHOR]

In this paper we study ensembles of finite real spectral triples equipped with a path integral over the space of possible Dirac operators. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a noncommutative analogue of integration over metrics, a key feature of a theory of quantum gravity. From these integrals in the so-called double scaling limit we derive critical exponents of minimal models from Liouville conformal field theory coupled with gravity. Additionally, the asymptotics of the partition function of these models satisfy differential equations such as Painlevé I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory. This is all proven using well-established and rigorous techniques from random matrix theory. [ABSTRACT FROM AUTHOR]

Unitary matrix integrals over symmetric polynomials play an important role in a wide variety of applications, including random matrix theory, gauge theory, number theory, and enumerative combinatorics. We derive novel results on such integrals and apply these and other identities to correlation functions of long-range random walks (LRRW) consisting of hard-core bosons. We generalize an identity due to Diaconis and Shahshahani which computes unitary matrix integrals over products of power sum polynomials. This allows us to derive two expressions for unitary matrix integrals over Schur polynomials, which can be directly applied to LRRW correlation functions. We then demonstrate a duality between distinct LRRW models, which we refer to as quasi-local particle-hole duality. We note a relation between the multiplication properties of power sum polynomials of degree n and fermionic particles hopping by n sites. This allows us to compute LRRW correlation functions in terms of auxiliary fermionic rather than hard-core bosonic systems. Inverting this reasoning leads to various results on long-range fermionic models as well. In principle, all results derived in this work can be implemented in experimental setups such as trapped ion systems, where LRRW models appear as an effective description. We further suggest specific correlation functions which may be applied to the benchmarking of such experimental setups. [ABSTRACT FROM AUTHOR]

We consider the jellium model of N particles on a line confined in an external harmonic potential and with a pairwise one-dimensional Coulomb repulsion of strength α > 0. Using a Coulomb gas method, we study the statistics of s = (1 / N) ∑ i = 1 N f (x i) where f (x), in principle, is an arbitrary smooth function. While the mean of s is easy to compute, the variance is nontrivial due to the long-range Coulomb interactions. In this paper we demonstrate that the fluctuations around this mean are Gaussian with a variance Var (s) ≈ b / N 3 for large N. In this paper, we provide an exact compact formula for the constant b = 1 / (4 α) ∫ − 2 α 2 α [ f ′ (x) ] 2 d x. In addition, we also calculate the full large deviation function characterizing the tails of the full distribution P (s , N) for several different examples of f (x). Our analytical predictions are confirmed by numerical simulations. [ABSTRACT FROM AUTHOR]

We study the time dynamics of random density matrices generated by evolving the same pure state using a Gaussian orthogonal ensemble (GOE) of Hamiltonians. We show that the spectral statistics of the resulting mixed state is well described by random matrix theory (RMT) and undergoes a crossover from the GOE to the Gaussian unitary ensemble (GUE) for short and large times respectively. Using a semi-analytical treatment relying on a power series of the density matrix as a function of time, we find that the crossover occurs in a characteristic time that scales as the inverse of the Hilbert space dimension. The RMT results are contrasted with a paradigmatic model of many-body localization in the chaotic regime, where the GUE statistics is reached at large times, while for short times the statistics strongly depends on the peculiarity of the considered subspace. [ABSTRACT FROM AUTHOR]

We review recent progress in the analytic study of random matrix models suggested by noncommutative geometry. One considers fuzzy spectral triples where the space of possible Dirac operators is assigned a probability distribution. These ensembles of Dirac operators are constructed as toy models of Euclidean quantum gravity on finite noncommutative spaces and display many interesting properties. The ensembles exhibit spectral phase transitions, and near these phase transitions they show manifold-like behavior. In certain cases one can recover Liouville quantum gravity in the double scaling limit. We highlight examples where bootstrap techniques, Coulomb gas methods, and Topological Recursion are applicable. [ABSTRACT FROM AUTHOR]

An overview over the microwave studies of chaotic systems is presented, performed by the authors and their co-workers in Marburg and Nice. In an historical overview the impact of Fritz Haake in particular in the beginning of the experiments is recognized. In the following sections two subjects are presented he was particularly interested in. One of them is the Bohigasâ€"Giannoniâ€"Schmit conjecture stating that the universal features of the spectra of chaotic systems are well described by random matrix theory. Microwave realizations of seven of the ten universal ensembles have been achieved, starting with the Gaussian orthogonal ensemble in the very first experiment, and ending with the chiral ensembles in a recent work. To do the measurements the systems have to be opened by attaching antennas to excite the microwaves. Antennas are theoretically taken into account in terms of a non-Hermitian effective Hamiltonian with an imaginary part taking care of the coupling to the environment. Results on level spacing and widths distribution in open systems are presented as well as on resonance trapping observed when changing the coupling to the environment. [ABSTRACT FROM AUTHOR]

We consider the problem of electronic quantum transport through ballistic mesoscopic systems with chaotic dynamics, connected to a three-terminal architecture in which one of the terminals has a tunnel barrier. Using a semiclassical approximation based on matrix integrals, we calculate several transport statistics, such as average and variance of conductance, average shot-noise power, among others, that give access to the extreme quantum regime (small channel numbers in the terminal) for broken and intact time-reversal symmetry, which the traditional random matrix approach does not access. As an application, we treat the dephasing regime. [ABSTRACT FROM AUTHOR]

Guztwiller’s trace formula is central to the semiclassical theory of quantum energy levels and spectral statistics in classically chaotic systems. Motivated by recent developments in random matrix theory and number theory, we elucidate a hierarchical structure in the way periodic orbits contribute to the trace formula that has implications for the value distribution of spectral determinants in quantum chaotic systems. [ABSTRACT FROM AUTHOR]

We apply the bootstrap technique to find the moments of certain multi-trace and multi-matrix random matrix models suggested by noncommutative geometry. Using bootstrapping we are able to find the relationships between the coupling constant of these models and their second moments. Using the Schwingerâ€"Dyson equations, all other moments can be expressed in terms of the coupling constant and the second moment. Explicit relations for higher mixed moments are also obtained. [ABSTRACT FROM AUTHOR]

The kicked top is one of the paradigmatic models in the study of quantum chaos (Haake et al 2018 Quantum Signatures of Chaos (Springer Series in Synergetics vol 54)). Recently it has been shown that the onset of quantum chaos in the kicked top can be related to the proliferation of Trotter errors in digital quantum simulation (DQS) of collective spin systems. Specifically, the proliferation of Trotter errors becomes manifest in expectation values of few-body observables strongly deviating from the target dynamics above a critical Trotter step, where the spectral statistics of the Floquet operator of the kicked top can be predicted by random matrix theory. In this work, we study these phenomena in the framework of Hamiltonian learning (HL). We show how a recently developed HL protocol can be employed to reconstruct the generator of the stroboscopic dynamics, i.e., the Floquet Hamiltonian, of the kicked top. We further show how the proliferation of Trotter errors is revealed by HL as the transition to a regime in which the dynamics cannot be approximately described by a low-order truncation of the Floquetâ€"Magnus expansion. This opens up new experimental possibilities for the analysis of Trotter errors on the level of the generator of the implemented dynamics, that can be generalized to the DQS of quantum many-body systems in a scalable way. This paper is in memory of our colleague and friend Fritz Haake. [ABSTRACT FROM AUTHOR]

The dependence of the chaotic phase of the Boseâ€"Hubbard Hamiltonian on particle number N, system size L and particle density is investigated in terms of spectral and eigenstate features. We analyse the development of the chaotic phase as the limit of infinite Hilbert space dimension is approached along different directions, and show that the fastest route to chaos is the path at fixed density n ≲ 1. The limit N â†' ∞ at constant L leads to a slower convergence of the chaotic phase towards the random matrix theory benchmarks. In this case, from the distribution of the eigenstate generalized fractal dimensions, the chaotic phase becomes more distinguishable from random matrix theory for larger N, in a similar way as along trajectories at fixed density. [ABSTRACT FROM AUTHOR]

In quantum chaos, the spectral statistics generally follows the predictions of random matrix theory (RMT). A notable exception is given by scar states, that enhance probability density around unstable periodic orbits of the classical system, therefore causing significant deviations of the spectral density from RMT expectations. In this work, the problem is considered of both RMT-ruled and scarred chaotic systems coupled to an opening. In particular, predictions are derived for the spectral density of a chaotic Hamiltonian scattering into a single- or multiple channels. The results are tested on paradigmatic quantum chaotic maps on a torus. The present report develops the intuitions previously sketched in Lippolis (2019 EuroPhys. Lett. 126 10003). [ABSTRACT FROM AUTHOR]

We report on experiments with a flat, superconducting microwave billiard with the shape of a quarter ellipse simulating a singular billiard, that is, a quantum billiard containing zero-range perturbations. The pointlike scatterers were realized with long antennas. Their coupling to the microwaves inside the cavity depends on frequency. A complete sequence of 1013 eigenfrequencies was identified rendering possible the investigation of spectral properties as function of frequency. They exhibit intermediate statistics and are well described by analytical results derived by Bogomolny, Gerland, Giraud and Schmit for singular billiards with shapes that generate an integrable classical dynamics. This comparison revealed a quadratic frequency dependence of the coupling parameter. The size of the chaotic component induced by the diffractive effects of the scatterers was determined by comparison with analytical results derived by Haake and Lenz for an additive random-matrix model, which interpolates between the models applicable for quantum systems with an integrable and chaotic classical dynamics, respectively. [ABSTRACT FROM AUTHOR]

We investigate the measurement-induced entanglement transition in quantum circuits built upon Dyson’s three circular ensembles (circular unitary, orthogonal, and symplectic ensembles; CUE, COE and CSE). We utilise the established model of a one-dimensional circuit evolving under alternating local random unitary gates and projective measurements performed with tunable rate, which for gates drawn from the CUE is known to display a transition from extensive to intensive entanglement scaling as the measurement rate is increased. By contrasting this case to the COE and CSE, we obtain insights into the interplay between the local entanglement generation by the gates and the entanglement reduction by the measurements. For this, we combine exact analytical random-matrix results for the entanglement generated by the individual gates in the different ensembles, and numerical results for the complete quantum circuit. These considerations include an efficient rephrasing of the statistical entangling power in terms of a characteristic entanglement matrix capturing the essence of Cartan’s KAK decomposition, and a general result for the eigenvalue statistics of antisymmetric matrices associated with the CSE. [ABSTRACT FROM AUTHOR]

We analyze statistical features of the ‘optimization landscape’ in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of a system of M linear equations in N unknowns: ( a k , x ) = b k , k = 1, …, M on the N -sphere x 2 = N. We treat both the N -component vectors a k and parameters b k as independent mean zero real Gaussian random variables. First, we derive the exact expressions for the mean number of stationary points of the least-square loss function in the overcomplete case M > N in the framework of the Kac-Rice approach combined with the random matrix theory for Wishart ensemble. Then we perform its asymptotic analysis as N â†' ∞ at a fixed α = M / N > 1 in various regimes. In particular, this analysis allows to extract the large deviation function for the density of the smallest Lagrange multiplier λ min associated with the problem, and in this way to find its most probable value. This can be further used to predict the asymptotic mean minimal value E min of the loss function as N â†' ∞. Finally, we develop an alternative approach based on the replica trick to conjecture the form of the large deviation function for the density of E min at N ≫ 1 and any fixed ratio α = M / N > 0. As a by-product, we find the compatibility threshold α c < 1 which is the value of α beyond which a large random linear system on the N -sphere becomes typically incompatible. [ABSTRACT FROM AUTHOR]

We study the statistical and dynamical properties of the quantum triangle map, whose classical counterpart can exhibit ergodic and mixing dynamics, but is never chaotic. Numerical results show that ergodicity is a sufficient condition for spectrum and eigenfunctions to follow the prediction of random matrix theory, even though the underlying classical dynamics is not chaotic. On the other hand, dynamical quantities such as the out-of-time-ordered correlator (OTOC) and the number of harmonics, exhibit a growth rate vanishing in the semiclassical limit, in agreement with the fact that classical dynamics has zero Lyapunov exponent. Our finding show that, while spectral statistics can be used to detect ergodicity, OTOC and number of harmonics are diagnostics of chaos. [ABSTRACT FROM AUTHOR]

In noninteracting limit, the density of states (dos) of a many body system can be expressed as a convolution of the single body dos of its subunits. We use the formulation to derive, in the edge of the spectrum, a differential equation for the ensemble averaged many body dos that is relatively easier to solve. Our analysis, based on the systems in which the subunits can be modelled by a Gaussian or Wishart random matrix ensemble, indicates that a rescaling of energy by the number of subunits leaves the many body dos in a mathematically invariant form. [ABSTRACT FROM AUTHOR]

The winding number is a concept in complex analysis which has, in the presence of chiral symmetry, a physics interpretation as the topological index belonging to gapped phases of fermions. We study statistical properties of this topological quantity. To this end, we set up a random matrix model for a chiral unitary system with a parametric dependence. We analytically calculate the discrete probability distribution of the winding numbers, as well as the parametric correlations functions of the winding number density. Moreover, we address aspects of universality for the two-point function of the winding number density by identifying a proper unfolding procedure. We conjecture the unfolded two-point function to be universal. [ABSTRACT FROM AUTHOR]

The local spectral statistics of random matrices forms distinct universality classes, strongly depending on the position in the spectrum. Surprisingly, the spacing between consecutive eigenvalues at the spectral edges has received little attention, where the density diverges or vanishes, respectively. This different behaviour is called hard or soft edge. We show that the spacings at the edges are almost indistinguishable from the spacing in the bulk of the spectrum. We present analytical results for consecutive spacings between the k th and (k + 1)st smallest eigenvalues in the chiral Gaussian unitary ensemble, both for finite- and large- n. The result depends on the number of the generic zero modes ν and the number of flavours N f, which are given in terms of characteristic polynomials, as motivated by quantum chromodynamics (QCD). We find that the convergence in n is very rapid. The same can be said separately about the limit k â†' ∞ (limit to the bulk) and ν â†' ∞ (limit to the soft edge). Interestingly, the Wigner surmise is a very good approximation for all these cases and, apart from k = 1, shows a deviation below one percent. These findings are corroborated with Monte-Carlo simulations. We finally compare for k = 1 with data from QCD on the lattice, being in this symmetry class. [ABSTRACT FROM AUTHOR]

The spectral moments of ensembles of sparse random block matrices are analytically evaluated in the limit of large order. The structure of the sparse matrix corresponds to the Erdösâ€"Renyi random graph. The blocks are i.i.d. random matrices of the classical ensembles GOE or GUE. The moments are evaluated for finite or infinite dimension of the blocks. The correspondences between sets of closed walks on trees and classes of irreducible partitions studied in free probability together with functional relations are powerful tools for analytic evaluation of the limiting moments. They are helpful to identify probability laws for the blocks and limits of the parameters which allow the evaluation of all the spectral moments and of the spectral density. [ABSTRACT FROM AUTHOR]

Building on the work of Fyodorov (2004) and Fyodorov and Nadal (2012) we examine the critical behaviour of population of saddles with fixed instability index k in high dimensional random energy landscapes. Such landscapes consist of a parabolic confining potential and a random part in N ≫ 1 dimensions. When the relative strength m of the parabolic part is decreasing below a critical value m c, the random energy landscapes exhibit a glass-like transition from a simple phase with very few critical points to a complex phase with the energy surface having exponentially many critical points. We obtain the annealed probability distribution of the instability index k by working out the mean size of the population of saddles with index k relative to the mean size of the entire population of critical points and observe toppling of stability hierarchy which accompanies the underlying glass-like transition. In the transition region m = m c + δN âˆ'1/2 the typical instability index scales as k = ÎşN 1/4 and the toppling mechanism affects whole instability index distribution, in particular the most probable value of Îş changes from Îş = 0 in the simple phase (δ > 0) to a non-zero value Îş max ∝ (âˆ' δ)3/2 in the complex phase (δ < 0). We also show that a similar phenomenon is observed in random landscapes with an additional fixed energy constraint and in the p -spin spherical model. [ABSTRACT FROM AUTHOR]

We consider a nonlinear autonomous random dynamical system of N degrees of freedom coupled by Gaussian random interactions and characterized by a continuous spectrum n ÎĽ (λ) of real positive relaxation rates. Using Kacâ€"Rice formalism, the computation of annealed complexities (both of stable equilibria and of all types of equilibria) is reduced to evaluating the averages involving the modulus of the determinant of the random Jacobian matrix. In the limit of large system N ≫ 1 we derive exact analytical results for the complexities for short-range correlated coupling fields, extending results previously obtained for the ‘homogeneous’ relaxation spectrum characterised by a single relaxation rate. We show the emergence of a ‘topology trivialisation’ transition from a complex phase with exponentially many equilibria to a simple phase with a single equilibrium as the magnitude of the random field is decreased. Within the complex phase the complexity of stable equilibria undergoes an additional transition from a phase with exponentially small probability to find a single stable equilibrium to a phase with exponentially many stable equilibria as the fraction of gradient component of the field is increased. The behaviour of the complexity at the transition is found only to depend on the small λ behaviour of the spectrum of relaxation rates n ÎĽ (λ) and thus conjectured to be universal. We also provide some insights into a counting problem motivated by a paper of Spivak and Zyuzin of 2004 about wave scattering in a disordered nonlinear medium. [ABSTRACT FROM AUTHOR]

The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wienerâ€"Hopf method it is demonstrated that quantum mechanics of a rectangular billiard with a barrier in the centre can be reduced to the investigation of a certain unitary matrix. Under heuristic assumptions this matrix is substituted by a special low-complexity random unitary matrix of independent interest. The main results of the paper are (i) spectral statistics of such billiards is insensitive to the barrier height and (ii) it is well described by the semi-Poisson distributions. [ABSTRACT FROM AUTHOR]

L -ensembles are a class of determinantal point processes which can be viewed as a statistical mechanical systems in the grand canonical ensemble. Circulant L -ensembles are the subclass which are locally translationally invariant and furthermore subject to periodic boundary conditions. Existing theory can very simply be specialised to this setting, allowing for the derivation of formulas for the system pressure, and the correlation kernel, in the thermodynamic limit. For a one-dimensional domain, this is possible when the circulant matrix is both real symmetric, or complex Hermitian. The special case of the former having a Gaussian functional form for the entries is shown to correspond to free fermions at finite temperature, and be generalisable to higher dimensions. A special case of the latter is shown to be the statistical mechanical model introduced by Gaudin to interpolate between Poisson and unitary symmetry statistics in random matrix theory. It is shown in all cases that the compressibility sum rule for the two-point correlation is obeyed, and the small and large distance asymptotics of the latter are considered. Also, a conjecture relating the asymptotic form of the hole probability to the pressure is verified. [ABSTRACT FROM AUTHOR]

Multi-parametric chiral random matrix ensembles are important tools to analyze the statistical behavior of generic complex systems with chiral symmetry. A recent study (Mondal and Shukla 2020 Phys. Rev. E 102 032131) of the former maps them to the chiral Brownian ensemble (Ch-BE) that appears as a non-equilibrium state of a single parametric crossover between two stationary chiral Hermitian ensembles. This motivates us to pursue a detailed statistical investigation of the spectral and strength fluctuations of the Ch-BE, with a focus on their behavior near zero energy region. The information can then be used for a wide range of complex systems with chiral symmetry. Our analysis also reveals connections of Ch-BE to generalized Calogero Sutherland Hamiltonian (CSH) and Wishart ensembles. This along with already known connections of complex systems without chirality to CSH strongly hints the later to be the 'backbone' Hamiltonian governing the spectral dynamics of complex systems. [ABSTRACT FROM AUTHOR]

An expansion of row Markov matrices in terms of matrices related to permutations with repetitions, is introduced. It generalises the Birkhoff–von Neumann expansion of doubly stochastic matrices in terms of permutation matrices (without repetitions). An interpretation of the formalism in terms of sequences of integers that open random safes described by the Markov matrices, is presented. Various quantities that describe probabilities and correlations in this context, are discussed. The Gini index is used to quantify the sparsity (certainty) of various probability vectors. The formalism is used in the context of multipartite quantum systems with finite dimensional Hilbert space, which can be viewed as quantum permutations with repetitions or as quantum safes. The scalar product of row Markov matrices, the various Gini indices, etc, are novel probabilistic quantities that describe the statistics of multipartite quantum systems. Local and global Fourier transforms are used to define locally dual and also globally dual statistical quantities. The latter depend on off-diagonal elements that entangle (in general) the various components of the system. Examples which demonstrate these ideas are also presented. [ABSTRACT FROM AUTHOR]

The bulk scaling limit of eigenvalue distribution on the complex plane of the complex Ginibre random matrices provides a determinantal point process (DPP). This point process is a typical example of disordered hyperuniform system characterized by an anomalous suppression of large-scale density fluctuations. As extensions of the Ginibre DPP, we consider a family of DPPs defined on the D-dimensional complex spaces , , in which the Ginibre DPP is realized when D = 1. This one-parameter family () of DPPs is called the Heisenberg family, since the correlation kernels are identified with the Szegő kernels for the reduced Heisenberg group. For each D, using the modified Bessel functions, an exact and useful expression is shown for the local number variance of points included in a ball with radius R in. We prove that any DPP in the Heisenberg family is in the hyperuniform state of class I, in the sense that the number variance behaves as R2D−1 as R → ∞. Our exact results provide asymptotic expansions of the number variances in large R. [ABSTRACT FROM AUTHOR]

We consider the problem of a semiclassical description of quantum chaotic transport, when a tunnel barrier is present in one of the leads. Using a semiclassical approach formulated in terms of a matrix model, we obtain transport moments as power series in the reflection probability of the barrier, whose coefficients are rational functions of the number of open channels M. Our results are therefore valid in the quantum regime and not only when M ≫ 1. The expressions we arrive at are not identical with the corresponding predictions from random matrix theory, but are in fact much simpler. Both theories agree as far as we can test. [ABSTRACT FROM AUTHOR]

Complex systems are often non-stationary, typical indicators are continuously changing statistical properties of time series. In particular, the correlations between different time series fluctuate. Models that describe the multivariate amplitude distributions of such systems are of considerable interest. Extending previous work, we view a set of measured, non-stationary correlation matrices as an ensemble for which we set up a random matrix model. We use this ensemble to average the stationary multivariate amplitude distributions measured on short time scales and thus obtain for large time scales multivariate amplitude distributions which feature heavy tails. We explicitly work out four cases, combining Gaussian and algebraic distributions. The results are either of closed forms or single integrals. We thus provide, first, explicit multivariate distributions for such non-stationary systems and, second, a tool that quantitatively captures the degree of non-stationarity in the correlations. [ABSTRACT FROM AUTHOR]

We present an analytic proof of the existence of phase transition in the large N limit of certain random noncommutative geometries. These geometries can be expressed as ensembles of Dirac operators. When they reduce to single matrix ensembles, one can apply the Coulomb gas method to find the empirical spectral distribution. We elaborate on the nature of the large N spectral distribution of the Dirac operator itself. Furthermore, we show that these models exhibit both a single and double cut region for certain values of the order parameter and find the exact value where the transition occurs. [ABSTRACT FROM AUTHOR]

This article is dedicated to the following class of problems. Start with an N × N Hermitian matrix randomly picked from a matrix ensemble—the reference matrix. Applying a rank-t perturbation to it, with t taking the values 1 ⩽ t ⩽ N, we study the difference between the spectra of the perturbed and the reference matrices as a function of t and its dependence on the underlying universality class of the random matrix ensemble. We consider both, the weaker kind of perturbation which either permutes or randomizes tdiagonal elements and a stronger perturbation randomizing successively trows and columns. In the first case we derive universal expressions in the scaled parameter τ = t/N for the expectation of the variance of the spectral shift functions, choosing as random-matrix ensembles Dyson's three Gaussian ensembles. In the second case we find an additional dependence on the matrix size N. [ABSTRACT FROM AUTHOR]

We develop a formalism to compute the statistics of the second largest eigenpair of weighted sparse graphs with N ≫ 1 nodes, finite mean connectivity and bounded maximal degree, in cases where the top eigenpair statistics is known. The problem can be cast in terms of optimisation of a quadratic form on the sphere with a fictitious temperature, after a suitable deflation of the original matrix model. We use the cavity and replica methods to find the solution in terms of self-consistent equations for auxiliary probability density functions, which can be solved by an improved population dynamics algorithm enforcing eigenvector orthogonality on-the-fly. The analytical results are in perfect agreement with numerical diagonalisation of large (weighted) adjacency matrices, focussing on the cases of random regular and Erdős–Rényi (ER) graphs. We further analyse the case of sparse Markov transition matrices for unbiased random walks, whose second largest eigenpair describes the non-equilibrium mode with the largest relaxation time. We also show that the population dynamics algorithm with population size NP does not actually capture the thermodynamic limit N → ∞ as commonly assumed: the accuracy of the population dynamics algorithm has a strongly non-monotonic behaviour as a function of NP, thus implying that an optimal size must be chosen to best reproduce the results from numerical diagonalisation of graphs of finite size N. [ABSTRACT FROM AUTHOR]

In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean. [ABSTRACT FROM AUTHOR]

Introducing projector-operator technique and algebra of Terwiel's cumulants we study stochastic linear partial differential equations with global and local disorder. We present the evolution equation for the mean-value of the field as a series in terms of Terwiel's cumulant operators. Then, we prove that if we use binary disorder with time exponential-correlated structure, as source of the stochastic perturbation, this series cuts leading to a treatable evolution equation. We apply this approach to find the exact mean-value solution of electromagnetic waves with stochastic absorption of energy in conducting media. This model shows the occurrence of novel time-scale separation phenomena. Local disorder in telegrapher's equation is also presented. Thus we show that strong disorder leads to anomalous behavior at short and long time regimes. In addition, other physical systems with global disorder are worked out to find exact mean-value solutions: finite-velocity diffusion in the presence of a deterministic force (Smoluchoswki-like process generalizing, in this way, Feynman–Kac's formula for its numerical solution); Lorentz' force on a fluctuating charge model (we calculate the diffusion coefficient transverse to the applied magnetic field); and a generalized non-Maxwellian velocity distribution (Ornstein–Uhlenbeck like process showing a noise-induced transition in the stationary distribution). [ABSTRACT FROM AUTHOR]

We study matrix integration over the classical Lie groups U(N), Sp(2N), SO(2N) and SO(2N + 1), using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz ± Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large N the partition functions, Wilson loops and Hopf links of Chern–Simons theory on S3 with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern–Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom. [ABSTRACT FROM AUTHOR]

We study generic open quantum systems with Markovian dissipation, focusing on a class of stochastic Liouvillian operators of Lindblad form with independent random dissipation channels (jump operators) and a random Hamiltonian. We establish that the global spectral features, the spectral gap, and the steady-state properties follow three different regimes as a function of the dissipation strength, whose boundaries depend on the particular quantity. Within each regime, we determine the scaling exponents with the dissipation strength and system size. We find that, for two or more dissipation channels, the spectral gap increases with the system size. The spectral distribution of the steady state is Poissonian at low dissipation strength and conforms to that of a random matrix once the dissipation is sufficiently strong. Our results can help to understand the long-time dynamics and steady-state properties of generic dissipative systems. [ABSTRACT FROM AUTHOR]

We analyze the dynamics of an algorithm for approximate inference with large Gaussian latent variable models in a student–teacher scenario. To model nontrivial dependencies between the latent variables, we assume random covariance matrices drawn from rotation invariant ensembles. For the case of perfect data-model matching, the knowledge of static order parameters derived from the replica method allows us to obtain efficient algorithmic updates in terms of matrix–vector multiplications with a fixed matrix. Using the dynamical functional approach, we obtain an exact effective stochastic process in the thermodynamic limit for a single node. From this, we obtain closed-form expressions for the rate of the convergence. Analytical results are in excellent agreement with simulations of single instances of large models. [ABSTRACT FROM AUTHOR]

Many solutions for scientific problems rely on finding the first (largest) eigenvalue and eigenvector of a particular matrix. We explore the distribution of the first eigenvector of a symmetric random sparse matrix. To analyze the properties of the first eigenvalue/vector, we employ a methodology based on the cavity method, a well-established technique in the statistical physics. A symmetric random sparse matrix in this paper can be regarded as an adjacency matrix for a network. We show that if a network is constructed by nodes that have two different types of degrees then the distribution of its eigenvector has fat tails such as the stable distribution (α < 2) under a certain condition; whereas if a network is constructed with nodes that have only one type of degree, the distribution of its first eigenvector becomes the Gaussian approximately. The method consisting of the cavity method and the population dynamical method clarifies these results. [ABSTRACT FROM AUTHOR]

With {Xi} independent N × N standard Gaussian random matrices, the probability that all eigenvalues are real for the matrix product Pm = XmXm − 1⋅⋅⋅X1 is expressed in terms of an N/2 × N/2 (N even) and (N + 1)/2 × (N + 1)/2 (N odd) determinant. The entries of the determinant are certain Meijer G-functions. In the case m = 2 high precision computation indicates that the entries are rational multiples of π2, with the denominator a power of 2, and that to leading order in N decays as . We are able to show that for general m and large N, with an explicit bm. An analytic demonstration that as m → ∞ is given. [ABSTRACT FROM AUTHOR]

Using a Coulomb gas technique, we compute analytically the probability that a large N × N Cauchy random matrix has N+ positive eigenvalues, where N+ is called the index of the ensemble. We show that this probability scales for large N as , where β is the Dyson index of the ensemble. The rate function ψC(κ) is computed in terms of single integrals that are easily evaluated numerically and amenable to an asymptotic analysis. We find that the rate function, around its minimum at κ = 1/2, has a quadratic behavior modulated by a logarithmic singularity. As a consequence, the variance of the index scales for large N as Var(N+) ∼ σCln N, where σC = 2/(βπ2) is twice as large as the corresponding prefactor in the Gaussian and Wishart cases. The analytical results are checked by numerical simulations and against an exact finite N formula which, for β = 2, can be derived using orthogonal polynomials. [ABSTRACT FROM AUTHOR]

Complex systems, and in particular random neural networks, are often described by randomly interacting dynamical systems with no specific symmetry. In that context, characterizing the number of relevant directions necessitates fine estimates on the Ginibre ensemble. In this fast track communication, we compute analytically the probability distribution of the number of eigenvalues NR with modulus greater than R (the index) of a large N _N random matrix in the real or complex Ginibre ensemble. We show that the fraction NR/N = p has a distribution scaling as exp(-βN2ΨR(p)) with β = 2 (respectively β = 1) for the complex (resp. real) Ginibre ensemble. For any p ∈ [0, 1], the equilibrium spectral densities as well as the rate function ΨR(p) are explicitly derived. This function displays a third order phase transition at the critical (minimum) value p* R = 1 - R², associated to a phase transition of the Coulomb gas. We deduce that, in the central regime, the fluctuations of the index NR around its typical value p*RN scale as N1/3. [ABSTRACT FROM AUTHOR]

In this paper we will first study the spectrum of certain large Euclidean random matrices. The entries of thesematrices are functions of n random points lp-norm uniformly sampled in N dimensional lp ball or lp sphere. Under the setting n/N → 0, as N and n tend to ∞, it is shown that the empirical distributions of eigenvalues of normalized Euclidean random matrices generated from the above two manifolds both converge weakly to semicircle law. We also investigate certain 'sample-covariance' type matrices whose entries are determined by n random positions from an N dimensional lp ellipsoid or its surface. When N → ∞and n → #8734;with N/n → 0, we prove that their empirical spectral distribution converge to the same fixed distribution. [ABSTRACT FROM AUTHOR]

We propose a matrix model which embodies the semiclassical approach to the problem of quantum transport in chaotic systems. Specifically, a matrix integral is presented whose perturbative expansion satisfies precisely the semiclassical diagrammatic rules for the calculation of general counting statistics. Evaluating it exactly, we show that it agrees with corresponding predictions from random matrix theory. This uncovers the algebraic structure behind the equivalence between these two approaches, and opens the way for further semiclassical calculations. [ABSTRACT FROM AUTHOR]