Large Deviation Function for the Number of Eigenvalues of Sparse Random Graphs Inside an Interval
Year: 2016
Type: article
Abstract: We present a general method to obtain the exact rate function ${\mathrm{\ensuremath{\Psi}}}_{[a,b]}(k)$ controlling the large deviation probability $\text{Prob}[{\mathcal{I}}_{N}[a,b]=kN]\ensuremath{\asymp}{e}^{\ensuremath{-}N{\mathrm{\ensuremath{\Psi}}}_{[a,b]}(k)}$ that an $N\ifmmode\times\else\texttimes\fi{}N$ 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 Erd\"os-R\'enyi 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 ${\ensuremath{\kappa}}_{2}/{\ensuremath{\kappa}}_{1}$ for the Anderson model on a regular graph satisfies $0<{\ensuremath{\kappa}}_{2}/{\ensuremath{\kappa}}_{1}<1$ 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.
Source: Physical Review Letters
Authors: Fernando L. Metz, Isaac Pérez Castillo
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2303721613