Circular Unitary Ensembles: Parametric Models and Their Asymptotic Maximum Likelihood Estimates

In this paper we compare two radar target direction-of-arrival (DOA) estimation algorithms, the classical moving window (MW) and the asymptotic maximum likelihood (AML) estimators. The first technique for azimuth DOA estimation exploits multiple detections in the same time-on-target and the second one exploits the fact that the radar antenna mechanical scanning impresses an amplitude modulation on the signals backscattered by the target. Performances of the estimators are numerically investigated through Monte Carlo simulation in terms of root-mean-square-error (RMSE), probability of detection for a fixed probability of false alarm, and probability of "splitting". The obtained results show that the asymptotic maximum likelihood estimator generally outperforms the classical moving window estimator.

Studies of density matrices for random quantum states lead naturally to the fixed trace Laguerre ensemble in random matrix theory. Previous studies have uncovered explicit rational function formulas for moments of purity statistic (trace of the squared density matrix), and also a third order linear differential equation satisfied by the eigenvalue density. We further probe the origin of these results from the viewpoint of integrability, which is taken here to mean wider classes of recursions and differential equations, and give extensions. Prominent in our study are first order linear matrix differential equations. One application given is to the derivation of the third order scalar equation for the density. Another is to obtain the explicit rational function formula for the variance of the purity statistic in the β generalised fixed trace Laguerre ensemble. In the original case (β = 2), the purity cumulants are expressed in terms of the large argument expansion of a particular σ-Painlevé IV transcendent. In a different but related direction, the exact computation of the two-point correlation for the fixed determinant circular unitary ensemble SU(N ) is given in the Appendix.Dedicated to the memory of Santosh Kumar and his work on exact results in RMT and their applications 1

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.

We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer "torsion." Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.

Optimal asymptotic least squares estimation in a singular set-up

Our current understanding of quantum chaos in many-body quantum systems hinges on the random matrix theory (RMT) behavior of eigenstates and their energy level statistics. Although RMT has been remarkably successful in describing "coarse" features of many-body quantum Hamiltonians in chaotic regimes, such as the Wigner-Dyson level spacing statistics or the volume-law behavior of eigenstate entanglement entropy, it remains a challenge to describe their "finer" features, particularly those arising from spatial locality. Here, we show that we can accurately describe the statistical behavior of eigenstate ensembles in many-body Hamiltonians by using pure random states with physical constraints that capture the essential features of the Hamiltonian, specifically spatial locality and symmetries. We demonstrate our approach on local spin Hamiltonians with a scalar U(1) charge. By constructing ensembles of constrained random states that account for two commuting scalar charges playing the role of energy and magnetization, we describe the patterns of entanglement of midspectrum eigenstates beyond their average volume-law behavior, including O(1) corrections and fluctuations, analytically and numerically. When defining the correspondence between quantum chaotic eigenstates in many-body Hamiltonians and RMT ensembles, our work highlights the important role played by spatial locality in describing universal features beyond the volume-law behavior.

Estimating the failure rate of the log-logistic distribution by smooth adaptive and bias-correction methods

Invariant ensembles, which are characterised by the joint distribution of eigenvalues P(λ 1, …, λ N ), play a central role in random matrix theory. We consider the truncated linear statistics LK=∑n=1Kf(λn) with 1 ⩽ K ⩽ N, λ 1 > λ 2 > ⋯ > λ N and f a given function. This quantity has been studied recently in the case where the function f is monotonous. Here, we consider the general case, where this function can be non-monotonous. Motivated by the physics of cold atoms, we study the example f(λ) = λ 2 in the Gaussian ensembles of random matrix theory. Using the Coulomb gas method, we obtain the distribution of the truncated linear statistics, in the limit N → ∞ and K → ∞, with κ = K/N fixed. We show that the distribution presents two essential singularities, which arise from two infinite order phase transitions for the underlying Coulomb gas. We further argue that this mechanism is universal, as it depends neither on the choice of the ensemble, nor on the function f.

We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these identities arise from combinatorial interpretations of certain properties of the $\beta$-Hermite and $\beta$-Laguerre ensembles of random matrix theory. We conclude by presenting two other identities obtained in the same way, for which finding combinatorial proofs is an open problem.

We study the level spacing distribution p(s) in the spectrum of random networks. According to our numerical results, the shape of p(s) in the Erds–Rényi (E–R) random graph is determined by the average degree ⟨k⟩ and p(s) undergoes a dramatic change when ⟨k⟩ is varied around the critical point of the percolation transition, ⟨k⟩ = 1. When ⟨k⟩ ≫ 1, the p(s) is described by the statistics of the Gaussian orthogonal ensemble (GOE), one of the major statistical ensembles in Random Matrix Theory, whereas at ⟨k⟩ = 1 it follows the Poisson level spacing distribution. Closely above the critical point, p(s) can be described in terms of an intermediate distribution between Poisson and the GOE, the Brody-distribution. Furthermore, below the critical point p(s) can be given with the help of the regularized Gamma-function. Motivated by these results, we analyse the behaviour of p(s) in real networks such as the internet, a word association network and a protein–protein interaction network as well. When the giant component of these networks is destroyed in a node deletion process simulating the networks subjected to intentional attack, their level spacing distribution undergoes a similar transition to that of the E–R graph.

We study the distribution of the largest eigenvalue in the "Pfaffian" classical ensembles of random matrix theory, namely in the Gaussian orthogonal (GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian" identities due to Knuth, we explicitly construct these semi-classical skew-orthogonal polynomials in terms of the semi-classical orthogonal polynomials studied by NM in the case of the GUE. With these polynomials we obtain expressions for the cumulative distribution functions of the largest eigenvalue in the GOE and the GSE. Further, by performing asymptotic analysis of these skew-orthogonal polynomials in the limit of large matrix size, we obtain an alternative derivation of the Tracy-Widom distributions for GOE and GSE. This asymptotic analysis relies on a certain Pfaffian identity, the proof of which employs the characterization of Pfaffians in terms of perfect matchings and link diagrams.

Estimation of polarimetric parameters has been a fundamental issue to assess biological tissues that have form birefringence or polarization scrambling in polarization-sensitive optical coherence tomography (PS-OCT). We present a mathematical framework to provide a maximum likelihood estimation of the target covariance matrix and its incoherent target decomposition to estimate a Jones matrix of a dominant scattering mechanism, called Cloude-Pottier decomposition, thereby deriving the phase retardation and the optic axis of the sample. In addition, we introduce entropy that shows the randomness of the polarization property. Underestimation of the entropy at a low sampling number is mitigated by asymptotic quasi maximum likelihood estimator. A bias of the entropy from random noises is corrected to show only the polarization property inherent in the sample. The theory is validated with experimental measurements of a glass plate and waveplates, and applied to the imaging of a healthy human eye anterior segment as an image filter.

Given a network of stations with incomplete and possibly imprecise inter-station range measurements, it is required to find the relative positions of the stations. Due to its asymptotic properties maximum likelihood estimation is discussed. Although the problem is quadratic, the proposed solution is based on solving a linear set of equations. For precise measurements we obtain explicitly the exact solution with a small number of operations. For noisy measurements the method provides an excellent initial point for the application of the Gerchberg-Saxton iterations. Proof of convergence is provided. The case of planar geometry is coached using complex numbers which reveals a strong relation to the celebrated problem of phase retrieval. We provide a compact, matrix form of the Cramer-Rao bound, small error analysis and evaluation of the computational load. Numerical examples are provided to corroborate the results.

Multifractality attributed to dual central limit-like convergence effects

We consider the spectral form factor of random unitary matrices as well as of Floquet matrices of kicked tops, as given by the (squared moduli of) the traces tn = tr Fn with the integer `time' n = 0,±1,±2, ... . For a typical matrix F the time dependence of the form factor |tn|2 looks erratic; only after a local time average over a suitably large time window n does a systematic time dependence become manifest. For matrices drawn from the circular unitary ensemble we prove ergodicity: in the limits of large matrix dimension and time window n the local time average has vanishingly small ensemble fluctuations and may be identified with the ensemble average. By numerically diagonalizing Floquet matrices of kicked tops with a globally chaotic classical limit we find the same ergodicity. As a byproduct we find that the traces tn of random matrices from the circular ensembles behave very much like independent Gaussian random numbers. Again, Floquet matrices of chaotic tops share that universal behaviour. It becomes clear that the form factor of chaotic dynamical systems can be fully faithful to random-matrix theory, not only in its locally time-averaged systematic time dependence but also in its fluctuations.