Local Density of States Correlations in the Lévy-Rosenzweig-Porter random matrix ensemble

We present an analytical calculation of the local density of states correlation function $\beta (\omega )$ in the Lévy-Rosenzweig-Porter random matrix ensemble at energy scales larger than the level spacing but smaller than the bandwidth. The only relevant energy scale in this limit is the typical level width ${\mathrm{\Gamma }}_0$. We show that $\beta (\omega \ll {\mathrm{\Gamma }}_0)\sim W\mathrm{/}{\mathrm{\Gamma }}_0$ (here $W$ is width of the band) whereas $\beta (\omega \gg {\mathrm{\Gamma }}_0)\sim (W\mathrm{/}{\mathrm{\Gamma }}_0)(\omega \mathrm{/}{\mathrm{\Gamma }}_0{)}^{-\mu }$ where $\mu$ is an index characterising the distribution of the matrix elements. We also provide an expression for the average return probability at long times: $\ln [R(t\gg {\mathrm{\Gamma }}_0^{-1})]\sim -({\mathrm{\Gamma }}_{0}t{)}^{\mu \mathrm{/}2}$. Numerical results based on the pool method and exact diagonalization are also provided and are in agreement with the analytical theory.