Statistical behavior of the characteristic polynomials of a family of pseudo-Hermitian Gaussian matrices

Abstract

In this paper, we extend previous studies conducted by the authors in a family of pseudo-Hermitian Gaussian matrices. Namely, we further the studies of the two pseudo-Hermitian random matrix cases previously considered, the first of a matrix of order N with two interacting blocks of sizes M and N − M and the second of a chessboard-like structured matrix of order N whose subdiagonals alternate between Hermiticity and pseudo-Hermiticity. Following an average characteristic polynomial approach, we obtain sequences of polynomials whose roots describe the average value of the polynomials of the matrices of the family at hand, for each case considered. We also present numerical results regarding the statistical behavior of the average characteristic polynomial, and contrast that to the spectral behavior of sample matrices of this family.

Pseudo-Hermitian $$\beta $$ β -Ensembles with Complex Eigenvalues

“…It is also noteworthy that the matrix η given in reference [24] which verifies (1) fluctuates around finite values for all its elements [25]. In particular, when N → ∞, the average goes as η N N → 1 and the variance goes as σ [26,27] A is one for which there exists a hermitian linear operator T such that its domain is the Hilbert space being considered, T is positive definite and bounded and, also, T A = A † T .…”

“…In the authors' previous work, the construction of a pseudo-Hermitian ensemble whose matrices are isospectral with those of the β-ensemble was presented [25]. This was done by assuming real tridiagonal non-Hermitian matrices with diagonal composed of random Gaussian variables of mean zero and variance one, lower subdiagonal equal to unity and kth element of the upper subdiagonal following the χ…”

“…To investigate this behavior under perturbations let us consider the case with parameter α = 1 from [25], that is…”

“…This non-Hermitian ensemble was constructed as an effort to provide an ensemble whose matrices could, in principle, model aspects of PT symmetric systems. More recently [25], it was shown that although matrices belonging to this non-Hermitian ensemble are pseudoHermitian, they also fit into the more restrictive category of quasi-Hermitian matrices such that no transition into the complex plane may be expected. Also in [25], a new ensemble was constructed, isospectral to the β-ensemble, whose eigenvalues transition into the complex plane under the effect of small perturbations.…”

“…In the following Sect. 3, the results of the isospectral matrices of [25] are reviewed and extended. Then in Sect.…”