Statistics of complex levels of random matrices for decaying systems

Non-hermitian random matrix theory: Method of hermitian reduction

“…This connection between eigenvalue distributions of complex matrices and two-dimensional electrostatics has been long known in the literature [3,5]. Continuing along this line, consider the Green's function G(z, z * ) in (2.10).…”

“…This formula has been known in the literature (see e.g., [3]), but there a seemingly unnatural small regulator had been introduced to make the argument of the logarithm in (2.9) strictly positive to avoid the branch point at 0. Our derivation of (2.6) suggests that that regulator is not really necessary.…”

“…It arises quite naturally in calculations involving Grassmann variables [2,3] where det (z − φ)(z * − φ † ) is the "fermion determinant". However, it involves the logarithm inside the average, as well as two derivatives.…”

“…However, it involves the logarithm inside the average, as well as two derivatives. Thus, unless one has a simple way of calculating the average in (2.9) (e.g., the replica method in the case of Gaussian ensembles [3]), (2.9) is not a practical expression for ρ(x, y). In most cases it is easier to calculate the Green's function…”

“…We will mention a few examples: QCD at finite chemical potential [2], nuclear decay [3], dissipative systems [3,4], and neural networks [5]. In all these examples, the random non-hermitean matrices were taken from a Gaussian distribution.…”