Spectral Form Factor K(t)

The spectral form factor K(t) = |Z(β+it)|²/Z(2β) probes eigenvalue correlations at all timescales. Random matrix theory predicts a universal dip-ramp-plateau structure: the ramp encodes level repulsion, the plateau signals Heisenberg time and information scrambling.

Matrix size N150

Ensemble samples30

β (energy filter)0.0

Ensemble type

K(t) averaged RMT prediction Disconnected (|⟨Z⟩|²)

The dip-ramp-plateau structure: (1) Early time: K(t) ~ 1, eigenvalues appear uncorrelated. (2) Dip: destructive interference from eigenvalue repulsion. (3) Linear ramp K(t) ~ t/t_H for GOE — a universal signature of chaotic level correlations. (4) Plateau at K=1 for t > t_H (Heisenberg time). The ramp is connected to information scrambling and the Page curve for black hole radiation.