Random antagonistic matrices (original) (raw)

On the spectrum of random anti-symmetric and tournament matrices

Random Matrices: Theory and Applications, 2016

We consider a discrete, non-Hermitian random matrix model, which can be expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We show that, asymptotically almost surely, the real parts of the eigenvalues of the non-Hermitian matrix around any fixed index are interlaced with those of the anti-symmetric matrix. Along the way, we show that some tools recently developed to study the eigenvalue distributions of Hermitian matrices extend to the anti-symmetric setting.

Mathematical Biology Random Leslie matrices in population dynamics

Journal of Mathematical Biology, 2011

We generalize the concept of the population growth rate when a Leslie matrix has random elements (correlated or not), i.e., characterizing the disorder in the vital parameters. In general, we present a perturbative formalism to deal with linear non-negative random matrix difference equations, then the non-trivial effective eigenvalue of which defines the long-time asymptotic dynamics of the mean-value population vector state is presented as the effective growth rate. This effective eigenvalue is calculated from the smallest positive root of a secular polynomial. Analytical (exact and perturbative calculations) results are presented for several models of disorder. In particular, a 3×3 numerical example is applied to study the effective growth rate characterizing the long-time dynamics of a biological population model. The present analysis is a perturbative method for finding the effective growth rate in cases when the vital parameters may have negative covariances across populations.

Random Leslie matrices in population dynamics

Journal of Mathematical Biology, 2011

We generalize the concept of the population growth rate when a Leslie matrix has random elements (correlated or not), i.e., characterizing the disorder in the vital parameters. In general, we present a perturbative formalism to deal with linear non-negative random matrix difference equations, then the non-trivial effective eigenvalue of which defines the long-time asymptotic dynamics of the mean-value population vector state is presented as the effective growth rate. This effective eigenvalue is calculated from the smallest positive root of a secular polynomial. Analytical (exact and perturbative calculations) results are presented for several models of disorder. In particular, a 3×3 numerical example is applied to study the effective growth rate characterizing the long-time dynamics of a biological population model. The present analysis is a perturbative method for finding the effective growth rate in cases when the vital parameters may have negative covariances across populations.

Dynamics on certain sets of stochastic matrices

Nonlinear Dynamics, 2011

We study iteration of polynomials on symmetric stochastic matrices. In particular, we focus on a certain one-parameter family of quadratic maps which exhibits chaotic behavior for a wide range of the parameters. The well-known dynamical behavior of the quadratic family on the interval, and its dependence on the parameter, is reproduced on the spectrum of the stochastic matrices. For certain subclasses of stochastic matrices the referred dynamical behavior is also obtained in the matrix entries. Since a stochastic matrix characterizes a Markov chain, we obtain a discrete dynamical system on the space of reversible Markov chains. Therefore, depending on the parameter, there are initial conditions for which the corresponding reversible Markov chains will lead under iteration to a fixed point, to a periodic point, or to an aperiodic point. Moreover, there are sensitivity to initial conditions and the coexistence of infinite repulsive periodic orbits, both features of chaos.

Random unistochastic matrices

Journal of Physics A: Mathematical and General, 2003

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 ), respectively). 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.

Nonlinear dynamics in constructing random bistochastic matrices

An efficient algorithm to generate a random bistochastic (doubly stochastic) matrix of a given size N is presented. The algorithm, based on alternating renormalization of rows and columns of a matrix, can be con-sidered as a nonlinear dynamical system in the space of matrices with non–negative elements. The Birkhoff polytope of bistochastic matrices thus can be considered as an attractor of this system. In the case N = 2 we derive explicit formulas for the probability distributions induced by the Dirichlet distribution in the set of stochastic matrices. For larger N we find the distribution which leads to a distribution locally flat at the center of the Birkhoff polytope. The probability density at this point allows us to arrive at an estimation of the volume of the Birkhoff polytope, consistent with recent asymptotic results.

On the Partial Connection Between Random Matrices and Interacting Particle Systems

2010

In the last decade there has been a lot of developments in the fields of random matrices, interacting particle systems, stochastic growth models, and the relations between these fields. For instance, several objects appearing in the limit of large matrices arise also in the long time limit for interacting particles and growth models. Examples are the famous Tracy-Widom distribution functions and the Airy 2 process.

Symmetric Pseudo-Random Matrices

IEEE Transactions on Information Theory, 2018

We consider the problem of generating symmetric pseudo-random sign (+ − 1) matrices based on the similarity of their spectra to Wigner's semicircular law. Using binary m-sequences (Golomb sequences) of lengths n = 2 m − 1, we give a simple explicit construction of circulant n × n sign matrices and show that their spectra converge to the semicircular law when n grows. The Kolmogorov complexity of the proposed matrices equals to that of Golomb sequences and is at most 2log 2 (n) bits.

Wishart and anti-Wishart random matrices

2003

We provide a compact exact representation for the distribution of the matrix elements of the Wishart-type random matrices A † A, for any finite number of rows and columns of A, 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 of reconstructing the redundant information hidden in Wishart matrices, with potential applications to numerous models based on biological, social and artificial intelligence networks. *