Spectral Form Factor — RMT Ramp & Plateau

Parameters

Matrix size N 50 Realizations 20 Smoothing η 0.5

SFF Features

Samples accumulated0

Matrix size N50

Plateau value K(∞)—

Ramp slope~t/N (GUE)

About

The spectral form factor K(t) = ⟨|Tr e^{iHt}|²⟩ probes level correlations at time scale t. For GUE/GOE (chaotic systems): early dip → linear ramp (t/N) → plateau at K=N. Poisson statistics (integrable systems) show no ramp. The ramp signals eigenvalue repulsion — a hallmark of quantum chaos. Used to diagnose chaos in black holes, many-body systems.