System-size dependence of the eigenvalue--eigenvector crossover mapping

Characterize the system-size dependence of the mapping between the eigenvalue-side crossover parameter and the eigenvector-side crossover parameter in random mixed graphs, including the mapping under alternative definitions of the eigenvalue crossover parameter.

Background

The paper compares an eigenvalue-side crossover parameter, extracted from first-order spacing-ratio fits, with an eigenvector-side parameter obtained from eigenvector-component distributions and multifractal dimensions. The comparison indicates that the standard Pandey--Mehta relation does not transfer directly to the sparse, discrete mixed-graph adjacency matrices: a model- and size-dependent prefactor is required, and the prefactor differs substantially between the studied system sizes. The authors therefore identify a need to determine how this mapping depends on system size and on the choice of eigenvalue statistic used to define the crossover parameter.