The spectrum of directed inhomogeneous random graphs follows a non-homogeneous circular law, with finite-rank outliers exhibiting explicit Gaussian fluctuations at scale sqrt(s_n/n).
The limiting spectral law for sparse iid matrices
3 Pith papers cite this work. Polarity classification is still indexing.
3
Pith papers citing it
abstract
Let $A$ be an $n\times n$ matrix with iid entries where $A_{ij} \sim \mathrm{Ber}(p)$ is a Bernoulli random variable with parameter $p = d/n$. We show that the empirical measure of the eigenvalues converges, in probability, to a deterministic distribution as $n \rightarrow \infty$. This essentially resolves a long line of work to determine the spectral laws of iid matrices and is the first known example for non-Hermitian random matrices at this level of sparsity.
fields
math.PR 3years
2026 3representative citing papers
Empirical spectral distributions of arbitrary-degree polynomials in Ginibre matrices converge to their Brown measures as matrix size tends to infinity.
Empirical spectral distribution of rescaled sparse row-regular 0-1 matrices converges in probability to the circular law when d = o(n) and d at least polylogarithmic.
citing papers explorer
-
Spectrum of Directed Inhomogeneous Random Graphs
The spectrum of directed inhomogeneous random graphs follows a non-homogeneous circular law, with finite-rank outliers exhibiting explicit Gaussian fluctuations at scale sqrt(s_n/n).
-
Brown measure convergence for the spectrum of polynomials in Ginibre matrices
Empirical spectral distributions of arbitrary-degree polynomials in Ginibre matrices converge to their Brown measures as matrix size tends to infinity.
-
The circular law for sparse random combinatorial matrices
Empirical spectral distribution of rescaled sparse row-regular 0-1 matrices converges in probability to the circular law when d = o(n) and d at least polylogarithmic.