Large deviation function for the number of eigenvalues of sparse random graphs inside an interval

Abstract

We present a general method to obtain the exact rate function ${\Psi }_{[a,b]}(k)$ controlling the large deviation probability $\text{Prob}[{\mathcal{I}}_N[a,b]=kN]\asymp e^{-N{\Psi }_{[a,b]}(k)}$ that a $N\times N$ 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 number variance ${\sigma }_N^2$ for the Anderson model on a regular graph scales as ${\sigma }_N^2\propto N$ ($N\gg 1$) for any nonzero disorder, which is consistent with the absence of level-repulsion in the extended phase. Our theoretical findings are thoroughly compared to numerical diagonalization in both cases, showing a reasonable good agreement.

6 pages, 3 figures

References in corpus (11)

• Many-Body Physics with Ultracold Gases
• The large deviation approach to statistical mechanics
• A random matrix model with localization and ergodic transitions
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Spectra of Sparse Random Matrices
• Quantum nonergodicity and fermion localization in a system with a single-particle mobility edge
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• On the localization transition in symmetric random matrices
• Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions
• Index statistical properties of sparse random graphs
• A unified approach for large deviations of bulk and extreme eigenvalues of the Wishart ensemble

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• Spectral Theory of Sparse Non-Hermitian Random Matrices
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• Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase
• The spectral density of dense random networks and the breakdown of the Wigner semicircle law
• Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs
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• Condensation of degrees emerging through a first-order phase transition in classical random graphs
• Dirty bosons on the Cayley tree: Bose-Einstein condensation versus ergodicity breaking
• Imaginary replica analysis of loopy regular random graphs
• A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles
• Analytic solution of the two-star model with correlated degrees
• Theory for the conditioned spectral density of non-invariant random matrices
• Replica-symmetric approach to the typical eigenvalue fluctuations of Gaussian random matrices
• Epidemic threshold and localization of the SIS model on directed complex networks
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• Phase transitions in atypical systems induced by a condensation transition on graphs
• The Wishart--Rosenzweig--Porter random matrix ensemble
• Top eigenpair statistics of diluted Wishart matrices
• Replica approach to the generalized Rosenzweig-Porter model
• Entanglement transitions induced by large deviations