Dissipative Spectral Form Factor of the Complex Elliptic Ginibre Ensemble across Various Non-Hermiticity Regimes

Abstract

We study the dissipative spectral form factor (DSFF) at complex time $T e^{iθ}$ for the complex elliptic Ginibre ensemble with non-Hermiticity parameter $τ\in [0,1)$. As the matrix dimension $N \to \infty$, we consider the natural scalings in both the time variable and the non-Hermiticity parameter, namely $T = O(N^γ)$ and $1 - τ= O(N^{-α})$. For all regimes $γ\ge 0$ and $α\ge 0$, we derive the precise asymptotic behaviour of both the disconnected and connected components of the DSFF. In particular, we explicitly characterise the dip--ramp--plateau structure, including the dip time and the Heisenberg time. In addition, we identify the mesoscopic regime $α\in (0,1)$, which interpolates between the behaviour of the DSFF of non-Hermitian random matrices and the spectral form factor (SFF) of Hermitian ensembles. We further provide an explicit description of the phase diagram, in which the ramp exhibits quadratic, linear, or intermediate behaviour depending on the scaling parameters.