Table of contents

In this introduction to the Journal of Physics A special issue on random matrix theory, we give a review of the main historical developments in random matrix theory. A short summary of the papers that appear in this special issue is also given.

In recent years there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications.

This paper presents some new statistical tests and new conjectures regarding the correspondence between the eigenvalues of random unitary matrices and the zeros of Riemann's zeta function. Global features such as the trace and number of eigenvalues in intervals are compared. Our results show satisfying match-ups between the two domains. They give examples of large natural datasets that follow classical distributions to high accuracy.

We calculate the discrete moments of the characteristic polynomial of a random unitary matrix, evaluated a small distance away from an eigenangle. Such results allow us to make conjectures about similar moments for the Riemann zeta function, and provide a uniform approach to understanding moments of the zeta function and its derivative.

We consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups. We compute their moments and find that the first few moments are Gaussian, whereas the limiting distribution is not. The precise number of Gaussian moments depends upon the particular statistic considered.

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are extended here to non-classical groups. We focus on an explicit example: the exceptional Lie group G₂. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G₂, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one-parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the seven-dimensional representation of G₂. The random matrix calculations extend to all exceptional Lie groups.

We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to be an accurate description for the horizontal distribution of the zeros of ζ'(s) to the right of the critical line. We show that as N → ∞ the fraction of the roots of Z'(U, z) that lie in the region 1 − x/(N − 1) ≤ |z| < 1 tends to a limit function. We derive asymptotic expressions for this function in the limits x → ∞ and x → 0 and compare them with numerical experiments.

Our interest is in the cumulative probabilities Pr(L(t) ≤ l)for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) ≤ l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik–Deift–Johansson and Baik–Rains.

The random orthogonal model (ROM) of Marinari–Parisi–Ritort [13, 14] is a model of statistical mechanics where the couplings among the spins are defined by a matrix chosen randomly within the orthogonal ensemble. It reproduces the most relevant properties of the Parisi solution of the Sherrington–Kirkpatrick model. Here we compute the energy distribution, and work out an estimate for the two-point correlation function. Moreover, we show an exponential increase with the system size of the number of metastable states also for non-zero magnetic field.

Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the attractive character of which determines the effectiveness of the algorithm. The behaviour of the difference map near fixed points is controlled by the relative orientation of the tangent spaces of the two constraint subspaces employed by the map. Since the dimensionalities involved are always large in practical applications, it is appropriate to use random matrix theory ideas to analyse the average-case convergence at fixed points. Optimal values of the γ parameters of the difference map are found which differ somewhat from the values previously obtained on the assumption of orthogonal tangent spaces.

Companies belonging to the same industrial branch are subject to similar economical influences. Hence, the time series of their stocks can show similar trends implying a correlation. Financial correlation matrices measure the unsystematic correlations between time series of stocks. Such information is important for risk management. It has been found by Laloux et al that the correlation matrices are 'noise dressed', a major reason being the finiteness of the time series. We present a new and alternative method to estimate this noise. We introduce a power mapping of the elements in the correlation matrix which suppresses the noise and thereby effectively 'prolongs' the time series. Neither further data processing nor additional input is needed. To develop and test our method, we use a model suggested by Noh which can be viewed as a special case of a 'factor model' in economics. We perform numerical simulations for the time series and obtain correlation matrices. We support the numerics by a qualitative analytical discussion. With our approach, different correlation structures buried under this noise can be detected. Our method is general and can be applied to all systems in which time series are measured.

We consider a generalization of the vicious walker model. Using a bijection map between the path configuration of the non-intersecting random walkers and the hook Young diagram, we compute the probability concerning the number of movements of the walker. Applying the saddle point method, we reveal that the scaling limit gives the Tracy–Widom distribution, which is the same with the limit distribution of the largest eigenvalues of the Gaussian unitary ensemble.

We consider the stochastic evolution of three variants of the RSK algorithm, giving both analytic descriptions and probabilistic interpretations. Symmetric functions play a key role, and the probabilistic interpretations are obtained by elementary Doob–Hunt theory. In each case, the evolution of the shape of the tableau obtained via the RSK algorithm can be interpreted as a conditioned random walk. This is intuitively appealing, and can be used for example to obtain certain relationships between orthogonal polynomial ensembles. In a certain scaling limit, there is a continous version of the RSK algorithm which inherits much of the structure exhibited in the discrete settings. Intertwining relationships between conditioned and unconditioned random walks are also given. In the continuous limit, these are related to the Harish-Chandra/Itzyksen–Zuber integral.

For one-matrix models with polynomial potentials, the explicit relationship between the partition function and the isomonodromic tau function for the 2 × 2 polynomial differential systems satisfied by the associated orthogonal polynomials is derived.

We study the critical behaviour of a random Hermitian one-matrix model with nonsymmetric interaction at a critical point, in which the eigenvalue density function has a zero of degree 2m, m ≥ 1, inside a cut. We prove that in the generic case, m = 1, the model exhibits a third-order phase transition in temperature. We formulate an ansatz for the double scaling limit of recurrence coefficients, which is consistent with the quasiperiodic asymptotics of recurrence coefficients in the low temperature region, and from this ansatz we derive the Painlevé II hierarchy of ordinary differential equations for the recurrence coefficients. In addition, we derive an integral kernel which governs the double scaling limit of correlation functions.

We study the Hermitian and normal two-matrix models in planar approximation for an arbitrary number of eigenvalue supports. Its planar graph interpretation is given. The study reveals a general structure of the underlying analytic complex curve, different from the hyperelliptic curve of the one-matrix model. The matrix model quantities are expressed through the periods of meromorphic generating differential on this curve and the partition function of the multiple support solution, as a function of filling numbers and coefficients of the matrix potential, is shown to be a quasiclassical tau-function. The relation to = 1 supersymmetric Yang–Mills theories is discussed. A general class of solvable multi-matrix models with tree-like interactions is considered.

The BC-type Calogero–Sutherland model (CSM) is an integrable extension of the ordinary A-type CSM that possesses a reflection symmetry point. The BC-CSM is related to the chiral classes of random matrix ensembles (RMEs) in exactly the same way as the A-CSM is related to the Dyson classes. We first develop the fermionic replica σ-model formalism suitable to treat all chiral RMEs. By exploiting 'generalized colour–flavour transformation' we then extend the method to find the exact asymptotics of the BC-CSM density profile. Consistency of our result with the c = 1 Gaussian conformal field theory description is verified. The emerging Friedel oscillations structure and sum rules are discussed in details. We also compute the distribution of the particle nearest to the reflection point.

In this paper, we review the method of constructing integrable deformations of the compactified c = 1 bosonic string theory by primary fields (momentum or winding modes), developed recently in collaboration with S Alexandrov and V Kazakov. The method is based on the formulation of the string theory as a matrix model. The flows generated by either momentum or winding modes (but not both) are integrable and satisfy the Toda lattice hierarchy.

The integral over the U(N) unitary group I = ∫ DU exp Tr AUBU† is re-examined. Various approaches and extensions are first reviewed. The second half of the paper deals with more recent developments: relation with integrable Toda lattice hierarchy, diagrammatic expansion and combinatorics, and what they teach us on the large N limit of log I.

Using the character expansion method, we generalize several well-known integrals over the unitary group to the case where general complex matrices appear in the integrand. These integrals are of interest in the theory of random matrices and may also find applications in lattice gauge theory.

We have found an exact formula expressing a general correlation function containing both products and ratios of characteristic polynomials of random Hermitian matrices. The answer is given in the form of a determinant. An essential difference from the previously studied correlation functions (of products only) is the appearance of non-polynomial functions along with the orthogonal polynomials. These non-polynomial functions are the Cauchy transforms of the orthogonal polynomials. The result is valid for arbitrary ensemble of β = 2 symmetry class and generalizes recent asymptotic formulae obtained for Gaussian unitary ensemble and its chiral counterpart by different methods.

Random matrix theory can be used to describe the transport properties of a chaotic quantum dot coupled to leads. In such a description, two approaches have been taken in the literature, considering either the Hamiltonian of the dot or its scattering matrix as the fundamental random quantity of the theory. In this paper, we calculate the first four moments of the distribution of the scattering matrix of a chaotic quantum dot with a time-dependent potential, thus establishing the foundations of a 'random scattering matrix approach' for time-dependent scattering. We consider the limit that the number of channels N coupling the quantum dot to the reservoirs is large. In this limit, the scattering matrix distribution is almost Gaussian, with small non-Gaussian corrections. Our results reproduce and unify results for conductance and pumped current previously obtained in the Hamiltonian approach. We also discuss an application to current noise.

We describe the singularities in the averaged density of states and the corresponding statistics of the energy levels in two- (2D) and three-dimensional (3D) chiral symmetric and time-reversal invariant disordered systems, realized in bipartite lattices with real off-diagonal disorder. For off-diagonal disorder of zero mean, we obtain a singular density of states in 2D which becomes much less pronounced in 3D, while the level-statistics can be described by a semi-Poisson distribution with mostly critical fractal states in 2D and Wigner surmise with mostly delocalized states in 3D. For logarithmic off-diagonal disorder of large strength, we find behaviour indistinguishable from ordinary disorder with strong localization in any dimension but in addition one-dimensional 1/|E| Dyson-like asymptotic spectral singularities. The off-diagonal disorder is also shown to enhance the propagation of two interacting particles similarly to systems with diagonal disorder. Although disordered models with chiral symmetry differ from non-chiral ones due to the presence of spectral singularities, both share the same qualitative localization properties except at the chiral symmetry point E = 0 which is critical.

The statistical properties of wavefunctions at the critical point of the spin quantum Hall transition are studied. The main emphasis is put on the determination of the spectrum of multifractal exponents Δ_q governing the scaling of moments ⟨|ψ|^2q}⟩ ∼ L^−qd−Δ_q with the system size L and the spatial decay of wavefunction correlations. Two- and three-point correlation functions are calculated analytically by means of mapping onto the classical percolation, yielding the values Δ₂ = −1/4 and Δ₃ = −3/4. The multifractality spectrum obtained from numerical simulations is given with a good accuracy by the parabolic approximation Δ_q ≃ q(1 − q)/8, but shows detectable deviations. We also study statistics of the two-point conductance g, in particular, the spectrum of exponents X_q characterizing the scaling of the moments ⟨g^q⟩. Relations between the spectra of critical exponents of wavefunctions (Δ_q), conductances (X_q) and Green functions at the localization transition with a critical density of states are discussed.

We study the transport properties of two electrons in a quasi one-dimensional disordered wire. The electrons are subject to both a disorder potential and a short range two-body interaction. Using the approach developed by Iida et al (1990 Ann. Phys., NY200 219), the supersymmetry technique and a suitable truncation of Hilbert space, we work out the two-point correlation function in the framework of a nonlinear σ model. We study the loop corrections to arbitrary order. We obtain a remarkably simple and physically transparent expression for the change of the localization length caused by the two-body interaction.

The scattering matrix was measured for microwave cavities with two antennae. It was analysed in the regime of overlapping resonances. The theoretical description in terms of a statistical scattering matrix and the rescaled Breit–Wigner approximation has been applied to this regime. The experimental results for the auto-correlation function show that the absorption in the cavity walls yields an exponential decay. This behaviour can only be modelled using a large number of weakly coupled channels. In comparison to the auto-correlation functions, the cross-correlation functions of the diagonal S-matrix elements display a more pronounced difference between regular and chaotic systems.

The paper discusses recent progress in understanding statistical properties of eigenvalues of (weakly) non-Hermitian and non-unitary random matrices. The first type of ensembles is of the form Ĵ = Ĥ − i, with Ĥ being a large random N × NHermitian matrix with independent entries 'deformed' by a certain anti-HermitianN × N matrix i satisfying in the limit of large dimension N the condition Tr Ĥ² ∝ N Tr ². Here can be either a random or just a fixed given Hermitian matrix. Ensembles of such a type with ≥ 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics. Related models are used to mimic complex spectra of the Dirac operator with chemical potential in the context of quantum chromodynamics.

Ensembles of the second type, arising naturally in scattering theory of discrete-time systems, are formed by N × N matrices  with complex entries such that † = Î − . For = 0 this coincides with the circular unitary ensemble, and 0 ≤ ≤ Î describes deviation from unitarity. Our result amounts to answering statistically the following old question: given the singular values of a matrix  describe the locus of its eigenvalues.

We systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non-trivial crossover from Wigner–Dyson statistics of real/unimodular eigenvalues typical of Hermitian/unitary matrices to Ginibre statistics in the complex plane typical of ensembles with strong non-Hermiticity: ⟨Tr Ĥ²⟩ ∝ ⟨Tr ²⟩ when N → ∞. Finally, we discuss (scarce) results available on eigenvector statistics for weakly non-Hermitian random matrices.

We present a random matrix theory for systems invariant under the joint action of parity, , and time reversal, , and, more generally, for pseudo-Hermitian systems. This brings out the appearance of the metric in a systematic way so that consistency with the postulates of quantum mechanics is maintained. Here we specialize only to 2 × 2 matrices and we construct a pseudo-unitary group. With explicit examples, nearest-neighbour level-spacing distributions for various classes of ensembles are found to exhibit a degree of level repulsion different from those hitherto known. This work is not only relevant to quantum chaos, but also to two-dimensional statistical mechanics and consistent non-local relativistic theories.

We describe in detail the solution of the extension of the chiral Gaussian unitary ensemble (chGUE) into the complex plane. The correlation functions of the model are first calculated for a finite number of N complex eigenvalues, where we exploit the existence of orthogonal Laguerre polynomials in the complex plane. When taking the large-N limit we derive new correlation functions in the case of weak and strong non-Hermiticity, thus describing the transition from the chGUE to a generalized Ginibre ensemble. We briefly discuss applications to the Dirac operator eigenvalue spectrum in quantum chromodynamics with non-vanishing chemical potential. This is an extended version of hep-th/0204068.

Comparison is made between the distribution of saddle points in the chaotic analytic function and in the characteristic polynomials of the Ginibre ensemble. Realizing the logarithmic derivative of these infinite polynomials as the electric field of a distribution of Coulombic charges at the zeros, a simple mean-field electrostatic argument shows that the density of saddles minus zeros falls off as π⁻¹ |z|⁻⁴ from the origin. This behaviour is expected to be general for finite or infinite polynomials with zeros uniformly randomly distributed in the complex plane, and which repel.

A non-Hermitian random matrix model proposed a few years ago has a remarkably intricate spectrum. Various attempts have been made to understand the spectrum, but even its dimension is not known. Using the Dyson–Schmidt equation, we show that the spectrum consists of a non-denumerable set of lines in the complex plane. Each line is the support of the spectrum of a periodic Hamiltonian, obtained by the infinite repetition of any finite sequence of the disorder variables. Our approach is based on the 'theory of words'. We make a complete study of all four-letter words. The spectrum is complicated because our matrix contains everything that will ever be written in the history of the universe, including this particular paper.

The study of the edge behaviour in the classical ensembles of Gaussian Hermitian matrices has led to the celebrated distributions of Tracy–Widom. Here we take up a similar line of inquiry in the non-Hermitian setting. We focus on the family of N × N random matrices with all entries independent and distributed as complex Gaussian of mean zero and variance 1/N. This is a fundamental non-Hermitian ensemble for which the eigenvalue density is known. Using this density, our main result is a limit law for the (scaled) spectral radius as N ↑ ∞. As a corollary, we get the analogous statement for the case where the complex Gaussians are replaced by quaternion Gaussians.

We compute the large scale (macroscopic) correlations in ensembles of normal random matrices with a general non-Gaussian measure and in ensembles of general non-Hermitian matrices with a class of non-Gaussian measures. In both cases, the eigenvalues are complex and in the large N limit they occupy a domain in the complex plane. For the case when the support of eigenvalues is a connected compact domain, we compute two-, three- and four-point connected correlation functions in the first non-vanishing order in 1/N, in a manner that the algorithm of computing higher correlations becomes clear. The correlation functions are expressed through the solution of the Dirichlet boundary problem in the domain complementary to the support of eigenvalues.

An ensemble of random unistochastic (orthostochastic) matrices is defined by taking squared moduli of elements of random unitary (orthogonal) matrices distributed according to the Haar measure on U(N) (or O(N)). An ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble. We study the distribution of complex eigenvalues of bistochastic, unistochastic and orthostochastic matrices in the complex plane. We compute averages (entropy, traces) over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices.

In quantum/wave systems with chaotic classical analogues, wavefunctions evolve in highly complex, yet deterministic ways. A slight perturbation of the system, though, will cause the evolution to diverge from its original behaviour increasingly with time. This divergence can be measured by the fidelity, which is defined as the squared overlap of the two time evolved states. For chaotic systems, two main decay regimes of either Gaussian or exponential behaviour have been identified depending on the strength of the perturbation. For perturbation strengths intermediate between the two regimes, the fidelity displays both forms of decay. By applying a complementary combination of random matrix and semiclassical theory, a uniform approximation can be derived that covers the full range of perturbation strengths. The time dependence is entirely fixed by the density of states and the so-called transition parameter, which can be related to the phase space volume of the system and the classical action diffusion constant, respectively. The accuracy of the approximations is illustrated with the standard map.

Maass waveforms of CM-type are a special kind of eigenfunction of the hyperbolic Laplacian whose 'defining' components (namely eigenvalue and Fourier coefficients) are given by simple formulae involving algebraic integers chosen from a suitable number field K/. In this paper, we report on some computer experiments aimed at ascertaining the extent to which the autocorrelation behaviour of CM-forms agrees with that of 'mock' (i.e. random) waveforms in the limit of high energy. Our results suggest that no significant differences are seen.

According to one of the basic conjectures in quantum chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms—'cat maps'. Our numerical experiments indicate that for 'generic' choices of cat maps, the unfolded consecutive spacing distribution in the irreducible components of the Nth quantization (given by the N-dimensional Weil representation) approaches the GOE/GSE law of random matrix theory. For certain special 'arithmetic' transformations, related to the Ramanujan graphs of Lubotzky, Phillips and Sarnak, the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.

We connect quantum graphs with infinite leads, and turn them into scattering systems. We show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics: typical poles, delay time and conductance distributions, Ericson fluctuations, and when considered statistically, the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles. The underlying classical dynamics can be defined, and it provides important parameters which are needed for the quantum theory. In particular, we derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. We use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields.

The correlations in the spectra of quantum systems are intimately related to correlations which are of genuine classical origin, and which appear in the spectra of actions of the classical periodic orbits of the corresponding classical systems. We review this duality and the semiclassical theory which brings it about. The conjecture that the quantum spectral statistics are described in terms of random matrix theory, leads to the proposition that the classical two-point correlation function is also given in terms of a universal function. We study in detail the spectrum of actions of the Baker map, and use it to illustrate the steps needed to reveal the classical correlations, their origin and their relation to symbolic dynamics.

We establish a general framework to explore parametric statistics of individual energy levels in unitary random matrix ensembles. For a generic confinement potential Tr W(H), we (i) find the joint distribution functions of the eigenvalues of H and H' = H + V for an arbitrary fixed V both for finite matrix size N and in the 'thermodynamic' N → ∞ limit; (ii) derive many-point parametric correlation functions of the two sets of eigenvalues and show that they are naturally parametrized by the eigenvalues of the reactance matrix for scattering off the 'potential' V; (iii) prove the universality of the correlation functions in unitary ensembles with non-Gaussian non-invariant confinement potential Tr W(H − V); (iv) establish a general scheme for exact calculation of level-number-dependent parametric correlation functions and apply the scheme to the calculation of intra-level velocity autocorrelation function and the distribution of parametric level shifts.

The embedded ensembles were introduced by Mon and French (1975 Ann. Phys., NY95 90) as physically more plausible stochastic models of many-body systems governed by one- and two-body interactions than provided by standard random-matrix theory. We review several approaches aimed at determining the spectral density, the spectral fluctuation properties and the ergodic properties of these ensembles: moments methods, numerical simulations, the replica trick, the eigenvector decomposition of the matrix of second moments and supersymmetry, the binary correlation approximation, and the study of correlations between matrix elements.

Distance matrices are matrices whose elements are the relative distances between points located on a certain manifold. In all cases considered here all their eigenvalues except one are non-positive. When the points are uncorrelated and randomly distributed we investigate the average density of their eigenvalues and the structure of their eigenfunctions. The spectrum exhibits delocalized and strongly localized states that possess different power-law average behaviour. The exponents depend only on the dimensionality of the manifold.

The Gaussian Penner matrix model is re-examined in the light of the results which have been found in double-well matrix models. The orthogonal polynomials for the Gaussian Penner model are shown to be the generalized Laguerre polynomials L^(α)_n(x) with α and x depending on N, the size of the matrix. An asymptotic formula for the orthogonal polynomials is derived following closely the orthogonal polynomial method of Deo (1997 Nucl. Phys. B 504 609). The universality found in the double-well matrix model is extended to include non-polynomial potentials. An asymptotic formula is also found for the Laguerre polynomial using the saddle-point method by rescaling α and x with N. Combining these results a novel asymptotic formula is found for the generalized Laguerre polynomials (different from that given in Szego's book) in a different asymptotic regime. This may have applications in mathematical and physical problems in the future. The density–density correlators are derived and are the same as those found for the double-well matrix models. These correlators in the smoothed large N limit are sensitive to odd and even N where N is the size of the matrix. These results for the two-point density–density correlation function may be useful in finding eigenvalue effects in experiments in mesoscopic systems or small metallic grains. There may be applications to string theory as well as the tunnelling of an eigenvalue from one valley to the other being an important quantity there.

We provide a compact exact representation for the distribution of the matrix elements of the Wishart-type random matrices ^†, for any finite number of rows and columns of , without any large N approximations. In particular, we treat the case when the Wishart-type random matrix contains redundant, non-random information, which is a new result. This representation is of interest for a procedure for reconstructing the redundant information hidden in Wishart matrices, with potential applications to numerous models based on biological, social and artificial intelligence networks.

We study energy level statistics of a critical random matrix ensemble of power-law banded complex Hermitian matrices. We compute the level compressibility via the level-number variance and compare it with the analytical formula for the exactly solvable model of Moshe, Neuberger and Shapiro.