General scheme for handling Hammerstein equations outside the spectral support

Develop a general method to treat the Hammerstein integral equations with Bessel kernels that determine the spectral densities of sparse Erdős–Rényi random graphs—specifically, the equations for adjacency matrices, ordinary graph Laplacians, and normalized graph Laplacians—in regions of the spectral parameter where the eigenvalue density is very small (outside the main support), for example by rearranging the ρ-integrals in the complex plane, so as to obtain reliable computations of g(ρ; z) and the associated spectral density in those regions.

Background

The paper proposes deterministic collocation-based numerical schemes to solve Hammerstein equations that arise from supersymmetric field-theoretic derivations of spectra for adjacency matrices and Laplacians of sparse Erdős–Rényi graphs. These methods perform well in the bulk of the spectrum for moderate to large average degree c, but deteriorate where the eigenvalue density becomes very small.

In such regions, the imaginary part of the saddle-point function g(ρ; z) grows slowly, leading to slowly decaying integrands and complicated oscillatory behavior in Bessel-kernel convolutions, which hampers polynomial-collocation approximations. The authors suggest that a reorganization of the integrals (e.g., contour deformation in the complex ρ-plane) could fix this, but report they do not yet have a working procedure.