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 P N (K,λ) that a large N × N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of P N (K,λ), 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 1 for |λ| > 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 nonmonotonic behavior as a function of λ. 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 λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.

Large-deviation theory for diluted Wishart random matrices

“…The peaks are commonly located at the eigenvalues of isolated, disconnected trees, that appear for small average degrees of the associated random graph model [34]. As clearly shown in figure 1(a), the presence of delta peaks in the spectrum manifests itself as discontinuities in the behavior of the first two cumulants, similarly to the analogous results derived for the adjacency matrix of random graphs [30,35].…”

Spectral theory of sparse non-Hermitian random matrices

“…In Figure 9 we present the spectral distribution ρ for the adjacency matrices of oriented Poisson graphs with mean indegree and outdegree c = 2 in the limit n → ∞ which results from solving (110), (112) and (113). In Figure 7 we compare direct diagonalisation results at finite size n with population dynamics results for n → ∞ and find a good agreement between both approaches.…”

“…The behaviour of various other spectral observables of non-Hermitian sparse random matrices have to our knowledge barely been studied, including the twopoint eigenvalue correlation function, the distribution of real eigenvalues, the localisation of eigenvectors [92,105], the study of the statistics of eigenvectors, the identification of contributions from the giant component and from finite clusters, the study of finite size corrections to the spectrum, and the computation of large deviation functions of spectral properties. In this regard, it would be very interesting to develop other methods for sparse non-Hermitian matrices such as the supersymmetric [106,107,108,109] or the replica method [110,111], which have been very successful to study properties of symmetric random matrices [112,113,114,115,116,117].…”

Theory for the conditioned spectral density of noninvariant random matrices

“…Here we state only the main results, while all details of our technique are discussed in the Supplemental Information [29]. Using the standard version of the replica method [30], combined with a representation of the index I N (x) in terms of complex logarithms [6,31], one rewrites the CSD as…”

“…(4) is a serious issue. We surmount this obstacle by following the replica approach as discussed in [31][32][33]. At first, these exponents are regarded as integer positive numbers, which allows to calculate the ensemble average and extract the large-N behavior of Q (N ) n .…”