Fig. 4

Covariance sizes for the Erdös–Rényi random graph ensemble. Pairwise and fourth order covariance sizes of the eigenvector components of the graph Laplacian for the Erdös–Rényi random graph ensemble. To evaluate the fourth moment and the mixed moments listed in the legend, we computed the average value over ≥100 independent samples for each value of n. Empirically, the expected value of $v_i{(l)}^4$ is approximately $\sqrt{2}{n}^{-5/3}$ (black); the dashed line is $\sqrt{2}{n}^{-5/3}$. The absolute value of the expectation of $v_i(l_1)v_j(l_2)$ is $n^{-2}$ if $i=j$ (blue) and essentially 0 if $i\ne j$ (data not shown; the average value was 10⁻¹⁹ or smaller). The expectation of $v_i{(l_1)}^2{v}_i{(l_2)}^2$ is approximately $n^{-2}$ (red). The absolute value of the expectation of $v_i{(l_1)}^2{v}_i(l_2)v_i(l_3)$ and $v_i(l_1)v_i(l_2)v_i(l_3)v_i(l_4)$ are both of order $n^{-3}$ (green and magenta). This is numerical evidence for assumptions A3–A5 below