Localization transition in symmetric random matrices

We study the behavior of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully connected Lévy matrices. We derive a critical line separating localized from extended states in the case of Lévy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.

Figure 1

Population dynamics results for the average DOS (solid lines) and IPR (dotted lines) for Laplacian matrices with $c=20$ and elements $\left\{K_l\right\}$ drawn from a Gaussian distribution. These results were obtained with $ϵ=0.001$ and a population of $5\times {10}^6$ samples for the distribution $W_{\lambda ,ϵ}\left(\omega \right)$. Numerical diagonalization results for the DOS $(\ast )$ and the IPR (○) obtained with an ensemble of 500 matrices of dimension $N=3000$ are shown. Error bars for the IPR are indicated. The insets show population dynamics results for the DOS (solid lines) and the IPR (dotted lines) as a function of $\lambda$ in the left and right tails of the spectrum of Laplacian matrices with $c=20$, $ϵ=0.001$ and fixed elements $\left\{K_l\right\}$.

Figure 2

Average IPR for Laplacian matrices as a function of $ϵ$ (population dynamics, top graph) and $N$ (numerical diagonalization, bottom graph) for $c=20$ and different numerical values of $\lambda$, which are shown explicitly on the figure. The elements $\left\{K_l\right\}$ are drawn from the Gaussian distribution described in the text. The diagonalization results were obtained considering an ensemble of 500 matrices for each $N$, with error bars included in this case. The population dynamics results were obtained with a population of $5\times {10}^6$ samples.

Figure 3

Comparison between numerical diagonalization (full lines) and population dynamics results (different types of symbols) for the average DOS of Lévy matrices with $\alpha =0.75(\ast )$ and $\alpha =1.25$ (○). The diagonalization results were obtained considering an ensemble of 1000 matrices of dimension $N=1500$. The population dynamics results were obtained considering $ϵ=0.001$, $\gamma =0.01$, and a population of ${10}^6$ samples.

Figure 4

Population dynamics results for the average IPR of Lévy matrices as a function of $\gamma$ for $\alpha =0.5$, $\lambda =1$ and a population of $5\times {10}^6$ samples. Three different values of $ϵ$ are shown: $ϵ=0.01$ (○), $ϵ=0.001$ (×) and $ϵ=0.0001$ $(\ast )$. The average IPR depends on $ϵ$ for $\gamma \to 0$.

Figure 5

Population dynamics results for the average IPR of Lévy matrices as a function of $\gamma$ for $\alpha =0.5$, $\lambda =5$ and a population of $5\times {10}^6$ samples. Three different values of $ϵ$ are shown: $ϵ=0.01$ (○), $ϵ=0.001$ (×), and $ϵ=0.0001$ $(\ast )$. The average IPR is independent of $ϵ$ for $\gamma \to 0$.

Figure 6

Population dynamics results for the critical line separating localized (L) from extended states (E) in the fully connected Lévy matrix. The results were obtained considering $ϵ=0.0001$, $\gamma =0.008$ and a population of ${10}^6$ samples.