Distribution Of Smallest Eigenvalue Research Articles

Heavy-tailed chiral random matrix theory

We study an unconventional chiral random matrix model with a heavy-tailed probabilistic weight. The model is shown to exhibit chiral symmetry breaking with no bilinear condensate, in analogy to the Stern phase of QCD. We solve the model analytically and obtain the microscopic spectral density and the smallest eigenvalue distribution for an arbitrary number of flavors and arbitrary quark masses. Exotic behaviors such as non-decoupling of heavy flavors and a power-law tail of the smallest eigenvalue distribution are illustrated.

The smallest eigenvalue distribution in the real Wishart–Laguerre ensemble with even topology

We consider rectangular random matrices of size belonging to the real Wishart–Laguerre ensemble also known as the chiral Gaussian orthogonal ensemble. This ensemble appears in many applications like QCD, mesoscopic physics, and time series analysis. We are particularly interested in the distribution of the smallest non-zero eigenvalue and the gap probability to find no eigenvalue in an interval . While for odd topology explicit closed results are known for finite and infinite matrix size, for even only recursive expressions in p are available. The smallest eigenvalue distribution as well as the gap probability for general even ν is equivalent to expectation values of characteristic polynomials raised to a half-integer power. The computation of such averages is done via a combination of skew-orthogonal polynomials and bosonisation methods. The results are given in terms of Pfaffian determinants both at finite p and in the hard edge scaling limit ( and ν fixed) for an arbitrary even topology ν. Numerical simulations for the correlated Wishart ensemble illustrate the universality of our results in this particular limit. These simulations point to a validity of the hard edge scaling limit beyond the invariant case.

Error probability analysis of a novel adaptive beamforming receiver for large‐scale multiple‐input–multiple‐output communication system

A novel adaptive beamforming (BF) receiver is proposed for a multiple-input–multiple-output communication system over multipath fading channels. By dividing a large number of receive antennas into many adaptive beamformers (ABFs), the proposed system reduces the array dimension in order to simplify the signal detection process and enjoys the benefit of BF techniques to improve the symbol-error-rate (SER) performance. The effectiveness of the proposed system depends on the number of ABFs and the number of antennas per ABF. The statistical properties of complex Wishart matrices are applied to analyse the SER performance of the proposed system with both maximum-likelihood (ML) detection and zero-forcing detection. Furthermore, the authors have derived an upper bound for SER of the ML detection based on the smallest eigenvalue distribution. Through the authors analysis, it has been shown that the proposed system using ML detection provides comparable SER performance, but with a lower complexity as compared with that of a conventional ML receiver because of reduced array dimensions.

Cooperative Spectrum Sensing Using Finite Demmel Condition Numbers

A novel and robust cooperative spectrum sensing scheme based on the exact distributions of Demmel Condition Number (DCN) of finite Wishart matrix is proposed in this paper. We also provide a new and simple method to determine the coefficient vector for the distribution of smallest eigenvalue, which is the key part in the generation of DCN distributions. A simple and exact expression of Cumulative Distribution Function of DCN for arbitrary matrix sizes is originally given to determine the theoretical threshold of the proposed spectrum sensing scheme.The simulations indicate that the proposed scheme can achieve better spectrum sensing performance comparing with conventional asymptotic methods based on infinite random matrix theory, and more importantly, the proposed algorithm is more robust against noise uncertainty.

Multi-Target Detection using Total Correlation for Noise Radar Systems

Target detection is one of the important functions of radar systems. In this paper, we present a detection method using total correlation (TC) based on information theory for noise radar systems which enables the detection of multiple targets at intermediate and low signal-to-noise ratio (SNR) regimes. The proposed method utilizes the largest eigenvalue of the sample covariance matrix to extract information from the transmitted signal replica, and outperforms the conventional TC detector when reflected signals have intermediate or low SNR values. Additionally, in order to avoid ambiguous target occurrence, we propose an adaptive threshold to guarantee the detection performance with the same receiving antenna elements for a given false alarm probability. The threshold is computed from the largest and smallest eigenvalue distributions based on random matrix theory. Simulations show that the proposed detection method can be used for a wide range of SNR environments, and the threshold provides definitive target detection.

Universality crossover between chiral random matrix ensembles and twisted SU(2) lattice Dirac spectra

Motivated by the statistical fluctuation of Dirac spectrum of QCD-like theories subjected to (pseudo)reality-violating perturbations and in the epsilon-regime, we compute the smallest eigenvalue distribution and the level spacing distribution of chiral and non-chiral parametric random matrix ensembles of Dyson-Mehta-Pandey type. To this end we employ the Nystrom method to numerically evaluate the Fredholm Pfaffian of the integral kernel for the chG(O,S)E-chGUE and G(O,S)E-GUE crossover. We confirm the validity and universality of our results by comparing them with several lattice models, namely fundamental and adjoint staggered Dirac spectra of SU(2) quenched lattice gauge theory under the twisted boundary condition (imaginary chemical potential) or perturbed by phase noise. Both in the zero-virtuality region and in the spectral bulk, excellent one-parameter fitting is achieved already on a small 4^4 lattice. Anticipated scaling of the fitting parameter with the twisting phase, mean level spacing, and the system size allows for precise determination of the pion decay (diffusion) constant F in the low-energy effective Lagrangian.