Density of states correlations in Lévy Rosenzweig-Porter model via supersymmetry approach

We studied global density-of-states correlation function R(\omega) R ( ω ) for Lévy-Rosenzweig-Porter random matrix ensemble in the non-ergodic extended phase. Using an extension of Efetov’s supersymmetry approach we calculated R(\omega) R ( ω ) exactly in all relevant ranges of \omega ω . At relatively low \omega ≤ \Gamma ω ≤ Γ (with \Gamma \gg \Delta Γ ≫ Δ being the effective miniband width) we found GUE-type oscillations with period of level spacing \Delta Δ , decaying exponentially at the Thouless energy scale E_{Th} = \sqrt{\Delta \Gamma/2\pi} E T h = Δ Γ / 2 π . At high energies \omega \gg E_{Th} ω ≫ E T h our results coincide with those obtained in [A. V. Lunkin and K. Tikhonov, SciPost Phys. 19, 015 (2025)] via cavity equation approach. Inverse of the effective miniband width, 1/\Gamma 1 / Γ , is shown to be given by the average of the local decay times over Lévy distribution.