Janossy Densities. I. Determinantal Ensembles

We derive an elementary formula for Janossy densities for determinantal point processes with a finite rank projection-type kernel. In particular, for =2 polynomial ensembles of random matrices we show that the Janossy densities on an interval I can be expressed in terms of the Christoffel–Darboux kernel for the orthogonal polynomials on the complement of I.     

Edge Universality for Orthogonal Ensembles of Random Matrices

We prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum. Our starting point is the result of Shcherbina (Commun. Math. Phys. 285, 957–974, 2009) on the representation of the reproducing matrix kernels of orthogonal ensembles in terms of scalar reproducing kernel of corresponding unitary ensemble.     

On the Relation Between Orthogonal,Symplectic and Unitary Matrix Ensembles

For the unitary ensembles of N×N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2×2 matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w/w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w/w. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations     

Duality of Orthogonal Polynomials on a Finite Set

We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways (using particles or holes) of computing the correlation functions of a discrete orthogonal polynomial ensemble.     

Correlations for parameter— dependent random matrices

Correlations for parameter-dependent Gaussian random matrices, intermediate between symmetric and Hermitian and antisymmetric Hermitian and Hermitian, are calculated. The (dynamical) density-density correlation between eigenvalues at different values of the parameter is calculated for the symmetric to Hermitian transition and the scaledN→∞ limit is computed. For the antisymmetric Hermitian to Hermitian transition the equal-parametern-point distribution function is calculated and the scaled limit computed. A circular version of the antisymmetric Hermitian to Hermitian transition is formulated. In the thermodynamic limit the equal-parameter distribution function is shown to coincide with the scaled-limit expression of this distribution for the Gaussian antisymmetric Hermitian to Hermitian transition. Furthermore, the thermodynamic limit of the corresponding density-density correlation is computed. The results for the correlations are illustrated by comparison with empirical correlations calculated from numerical data obtained from computer-generated Gaussian random matrices.     

Random matrices and matrix models: The JNU lectures

A course of lectures was given at the Jawaharlal Nehru University and the Jamia Milia Islamia, New Delhi, during February—March 1996. The following notes were distributed to the audience before each lecture. These notes, which are sketchy and do not go in details, were meant to help students follow the standard literature on the subject. They are collected here (with the exercises!) in the hope that they might prove useful to a larger community of research workers.     

Linear Statistics of Point Processes via Orthogonal Polynomials

For arbitrary β>0, we use the orthogonal polynomials techniques developed in (Killip and Nenciu in , 2005; Killip and Nenciu in Int. Math. Res. Not. 50: 2665–2701, 2004) to study certain linear statistics associated with the circular and Jacobi β ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.     

Universality for Orthogonal and Symplectic Laguerre-Type Ensembles

We give a proof of the Universality Conjecture for orthogonal (β=1) and symplectic (β=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels K _{n, β}, correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (β=2) Laguerre-type ensembles have been proved by the fourth author in Ref. 23. The varying weight case at the hard spectral edge was analyzed in Ref. 13 for β=2: In this paper we do not consider varying weights. Our proof follows closely the work of the first two authors who showed in Refs. 7, 8 analogous results for Hermite-type ensembles. As in Refs. 7, 8 we use the version of the orthogonal polynomial method presented in Refs. 22, 25, to analyze the local eigenvalue statistics. The necessary asymptotic information on the Laguerre-type orthogonal polynomials is taken from Ref. 23.     

On the multifractal analysis of Bernoulli convolutions. II. Dimensions

We show how the formalism developed in a previous paper allows us to exhibit the multifractal nature of the infinitely convolved Bernoulli measures for the golden mean. In this second part we show how the Hausdorff dimension of the set where the measure has a power law singularity of strength is related to the large-deviation function given in Part I.     

Classical Skew Orthogonal Polynomials and Random Matrices

Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.     

On the Proof of Universality for Orthogonal and Symplectic Ensembles in Random Matrix Theory

We give a streamlined proof of a quantitative version of a result from P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) which is crucial for the proof of universality in the bulk P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) and also at the edge P. Deift and D. Gioev, {Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices. Comm. Pure Appl. Math. (in press) for orthogonal and symplectic ensembles of random matrices. As a byproduct, this result gives asymptotic information on a certain ratio of the β=1,2,4 partition functions for log gases.     

Vicious random walkers in the limit of a large number of walkers

The vicious random walker problem on a line is studied in the limit of a large number of walkers. The multidimensional integral representing the probability that thep walkers will survive a timet (denotedP _t ^(p) ) is shown to be analogous to the partition function of a particular one-component Coulomb gas. By assuming the existence of the thermodynamic limit for the Coulomb gas, one can deduce asymptotic formulas forP _t ^(p) in the large-p, large-t limit. A straightforward analysis gives rigorous asymptotic formulas for the probability that after a timet the walkers are in their initial configuration (this event is termed a reunion). Consequently, asymptotic formulas for the conditional probability of a reunion, given that all walkers survive, are derived. Also, an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in timet, given that allp walkers survive, is calculated in the limittp.     

Grand Canonical Ensembles of Sparse Networks and Bayesian Inference

Maximum entropy network ensembles have been very successful in modelling sparse network topologies and in solving challenging inference problems. However the sparse maximum entropy network models proposed so far have fixed number of nodes and are typically not exchangeable. Here we consider hierarchical models for exchangeable networks in the sparse limit, i.e., with the total number of links scaling linearly with the total number of nodes. The approach is grand canonical, i.e., the number of nodes of the network is not fixed a priori: it is finite but can be arbitrarily large. In this way the grand canonical network ensembles circumvent the difficulties in treating infinite sparse exchangeable networks which according to the Aldous-Hoover theorem must vanish. The approach can treat networks with given degree distribution or networks with given distribution of latent variables. When only a subgraph induced by a subset of nodes is known, this model allows a Bayesian estimation of the network size and the degree sequence (or the sequence of latent variables) of the entire network which can be used for network reconstruction.     

Pre- and Post-Selected Ensembles and Time-Symmetry in Quantum Mechanics

It has recently been argued (Shimony, Erkenntnis 45:337, 1997) that time-symmetry does not hold for pre- and post-selected ensembles in quantum mechanics. That conclusion depends on what is meant by “time-symmetry” in relation to those types of ensembles. It is shown that on the conventional view of time-symmetry, pre- and post-selected ensembles are time-symmetric as was originally proposed.     

On the multifractal analysis of Bernoulli convolutions. I. Large-deviation results

We show how the formalism developed in a previous paper allows us to exhibit the multifractal nature of the infinitely convolved Bernoulli measures , for the golden mean. In this first part we establish some large-deviation results for random products of matrices, using perturbation theory of quasicompact operators.     

On the joint eigenvalue distribution for the matrix ensembles with non zero mean

Exact distributions are given for the two-dimensional case when the mean of the off-diagonal element is non-zero. The joint eigenvalue distribution for theN dimensional case, derived using the volume element in the space ofN ×N orthogonal matrices, is checked by rederiving the exact results forN=2. The smooth nature of theN-dimensional joint distribution supports the claim of the method of moments that the single eigenvalue distribution is a smooth function of the ratio of mean-to-mean square deviation.     

Gaussian-Random Ensembles of Pseudo-Hermitian Matrices

Attention has been brought to the possibility that statistical fluctuation properties of several complex spectra, or well-known number sequences, may display strong signatures that the Hamiltonian yielding them as eigenvalues is PT-symmetric (pseudo-Hermitian). We find that the random matrix theory of pseudo-Hermitian Hamiltonians gives rise to new universalities of level-spacing distributions other than those of GOE, GUE and GSE of Wigner and Dyson. We call the new proposals as Gaussian Pseudo-Orthogonal Ensemble and Gaussian Pseudo-Unitary Ensemble. We are also led to speculate that the enigmatic Riemann-zeros (1/2±it _n would rather correspond to some PT-symmetric (pseudo-Hermitian) Hamiltonian.     

Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential

In this work, we develop an orthogonal-polynomials approach for random matrices with orthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, which are a generalization of Gaussian random matrices. The representation of the correlation functions in these matrix models, via the technique of quaternion determinants, makes use of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices, which essentially express them in terms of the reproducing kernel of the theory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given by Bleher and Its for the corresponding orthogonal polynomials.     

No-relationship Between Impossibility of Faster-Than-Light Quantum Communication and Distinction of Ensembles with the Same Density Matrix

It has been claimed in the literature that impossibility of faster-than-light quantum communication has an origin of indistinguishability of ensembles with the same density matrix. We show that the two concepts are not related.We argue that even with an ideal single-atom-precision measurement, it is generally impossible to produce two ensembles with exactly the same density matrix; or to produce ensembles with the same density matrix, classical communication is necessary. Hence the impossibility of faster-than-light communication does not imply the indistinguishability of ensembles with the same density matrix.