Large Deviation Function for the Number of Eigenvalues of Sparse Random Graphs Inside an Interval

We present a general method to obtain the exact rate function ${\mathrm{\Psi }}_{[a,b]}(k)$ controlling the large deviation probability $\text{Prob}[{\mathcal{I}}_N[a,b]=kN]\asymp e^{-N{\mathrm{\Psi }}_{[a,b]}(k)}$ that an $N\times N$ sparse random matrix has ${\mathcal{I}}_N[a,b]=kN$ eigenvalues inside the interval $[a,b]$. The method is applied to study the eigenvalue statistics in two distinct examples: (i) the shifted index number of eigenvalues for an ensemble of Erdös-Rényi graphs and (ii) the number of eigenvalues within a bounded region of the spectrum for the Anderson model on regular random graphs. A salient feature of the rate function in both cases is that, unlike rotationally invariant random matrices, it is asymmetric with respect to its minimum. The asymmetric character depends on the disorder in a way that is compatible with the distinct eigenvalue statistics corresponding to localized and delocalized eigenstates. The results also show that the level compressibility ${\kappa }_2/{\kappa }_1$ for the Anderson model on a regular graph satisfies $0<{\kappa }_2/{\kappa }_1<1$ in the bulk regime, in contrast with the behavior found in Gaussian random matrices. Our theoretical findings are thoroughly compared to numerical diagonalization in both cases, showing a reasonable good agreement.

Figure 1

Rate function ${\mathrm{\Psi }}_L^{(p)}(k)$ of the fraction of eigenvalues inside $(-\infty ,L]$ for Erdös-Rényi graphs with $L=-1$ and average connectivity $c=3$. The solid line is the population dynamics results and the symbols correspond to numerical diagonalization of matrices of sizes $N=50$ (yellow pentagons), $N=100$ (orange rhombic symbols), and $N=300$ (dark-red triangles), using ensembles with $7\times 10^9$, $6\times 10^8$, and $2\times 10^8$ samples, respectively.

Figure 2

Population dynamics results for the rate function ${\mathrm{\Psi }}_L^{(p)}(k)$ of the fraction of eigenvalues inside $(-\infty ,L]$ for Erdös-Rényi graphs with $L=-1$ and different values of the average connectivity $c$. The typical value $k_{\text{typ}}$ of the shifted index is defined in the main text.

Figure 3

Population dynamics results for the rate function ${\mathrm{\Psi }}_L^{(a)}(k)$ of the fraction of eigenvalues inside $[-L,L]$ for the Anderson model on a regular random graph with fixed connectivity $c=3$, $L=1$ and different disorder strengths $W$. The solid red line is the rate function of a binomial distribution, where $2L/W$ is the probability that an eigenvalue falls into $[-L,L]$.

Figure 4

Populations dynamics results (solid lines) for the cumulant ratio ${\kappa }_2/{\kappa }_1$ of the number of eigenvalues within $[-L,L]$ for the Anderson model on a regular random graph with connectivity $c=3$ and different $L$: $L=1/2$ (dark red), $L=1$ (yellow) and $L=2$ (blue). We also present numerical diagonalization results (symbols) for matrices of size $N=1000$ and average over $5\times 10^3$ samples, for $L=1/2$ and $L=1$, and over $10^4$ samples for $L=2$. The shaded area around each curve represents the error bars.