Wigner Surmise Research Articles

Random matrix statistics and safety rest areas on interstates in the United States

We analyze physical spacings between locations of safety rest areas on interstates in the United States. We show normalized safety rest area spacings on major interstates exhibit Wigner surmise (WS) statistics, which align with the eigenvalue spacings for the Gaussian unitary ensemble from random matrix theory as well as the one-dimensional gas interactions via the Coulomb potential. We identify economic and geographic regional traits at the state level that exhibit Poissonian statistics, which become more pronounced with increased geographical obstacles in interstate travel. Other regional filters (e.g. historical or political) produced results that did not diverge substantially from the overall WS model.

Rayleigh's Distribution, Wigner's Surmise and Equation of the Diffusion

Non-Gaussian distributions occur in systems that do not follow strictly the prescriptions of standard statistics. Important example of non-Gaussian statistics is distribution which was introduced by Lord Rayleigh in connection with the problem of interference of harmonic oscillations with random phases [1, 2]. This distribution is known also as Wigner's surmise distribution giving a remarkably good description of the level repulsion observed in neutron scattering [3, 4]. As the mathematical form of both is identical, we call them Rayleigh Wigner's distribution (RWD). Rayleigh Wigner's distribution appears in computation of the large zeros of Riemann's zeta function on the critical line, which according to the Montgomery Odlyzko law have the same statistical properties as the distribution of eigenvalue spacings in a Gaussian unitary ensemble [5 9]. Rayleigh Wigner's distribution is also met in social sciences; e.g. the bus system in Cuernavaca, Mexico, is subject to this distribution [10]. This wide range of applications of RWD is a new illustration of Wigner's opinion on unreasonable e ectiveness of mathematics [11].

Wigner surmises and the two-dimensional Poisson-Voronoi tessellation

Consider the set of points defined by the intersection of a one-dimensional (random) line and a Voronoi tessellation generated by a two-dimensional homogeneous Poisson point process. We show that the nearest-neighbor and higher-order spacing statistics between this set of points and the Poisson points that were used to generate the Voronoi tessellation are given by the Wigner surmises of random matrix theory (with even-integer values of the level-repulsion parameter), when appropriate normalizations are implemented.

Transitional Wigner surmises from the spacing distribution of 4 × 4 matrices

Recently Nieminen introduced in this journal a 4 × 4 random matrix to study transitional distributions between Wigner surmises of random matrix theory. We find analytical expressions for the distributions that they obtained numerically. We also study the Ginibre-to-GOE transition.

Localization–delocalization transition in two-dimensional system with correlated disorder

We study the localization properties of the two-dimensional (2D) Anderson model with diagonal correlated disorder named visible vw and non-visible nw sites. Using the exact diagonalization of Anderson Hamiltonian, investigating the behavior of the density of states (DOS), we found that states in the center of the band E = 0 show critical behavior. The energy levels statistics (ELS) is also examined, we found a crossover of the nearest-neighbor level spacing distribution P ( s ) from Wigner Surmise distribution at small correlated disorder vw and nw (indicating a metallic behavior) to the Poisson distribution at large correlated disorder vw and nw characteristic for localized states. In addition, an analysis of width M dependence of the reduced localization length λ / M is obtained by the transfer-matrix method (TMM). For all investigated strengths of correlated disorder at E = 0 and E =+ 2 , the reduced localization length λ / M always decreases with increasing width. We found only delocalized states in suggested energy and disorder range.

Eigenvalue spacing statistics of a four-matrix model of some four-by-four random matrices

A simple four-matrix model consisting of some 4 × 4 real and imaginary random matrices and three parameters is introduced. It is shown that the eigenvalue spacing statistics of an ensemble of such matrices can be used to describe transitions between all the Wigner surmises of random matrix theory. Formulae for the nearest-neighbour spacing distributions of the ensemble are given for various parameter values.

Wigner surmises and the two-dimensional homogeneous Poisson point process

We derive a set of identities that relate the higher-order interpoint spacing statistics of the two-dimensional homogeneous Poisson point process to the Wigner surmises for the higher-order spacing distributions of eigenvalues from the three classical random matrix ensembles. We also report a remarkable identity that equates the second-nearest-neighbor spacing statistics of the points of the Poisson process and the nearest-neighbor spacing statistics of complex eigenvalues from Ginibre's ensemble of 2 x 2 complex non-Hermitian random matrices.

Non-extensive random matrix theory—a bridge connecting chaotic and regular dynamics

We consider a possible generalization of the random matrix theory, which involves the maximization of Tsallis' q-parametrized entropy. We discuss the dependence of the spacing distribution on q using a non-extensive generalization of Wigner's surmises for ensembles belonging to the orthogonal, unitary and symplectic symmetry universal classes.