Index statistical properties of sparse random graphs

Abstract

Using the replica method, we develop an analytical approach to compute the characteristic function for the probability ${\mathcal{P}}_N(K,\lambda )$ that a large $N\times N$ adjacency matrix of sparse random graphs has $K$ eigenvalues below a threshold $\lambda$. The method allows to determine, in principle, all moments of ${\mathcal{P}}_N(K,\lambda )$, from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with $N\gg 1$ for $|\lambda |>0$, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of $\lambda$. These results contrast with rotationally invariant random matrices, where the index variance scales only as $\ln N$, with an universal prefactor that is independent of $\lambda$. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.

10 pages, 5 figures

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Cited by in corpus (11)

• Anderson localization on random regular graphs
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase
• Large deviation function for the number of eigenvalues of sparse random graphs inside an interval
• Large deviation theory for diluted Wishart random matrices
• Imaginary replica analysis of loopy regular random graphs
• Theory for the conditioned spectral density of non-invariant random matrices
• Replica-symmetric approach to the typical eigenvalue fluctuations of Gaussian random matrices
• Index of a matrix, complex logarithms, and multidimensional Fresnel integrals
• Analytic approach for the number statistics of non-Hermitian random matrices
• Top eigenpair statistics of diluted Wishart matrices