Spectra of sparse non-hermitian random matrices: an analytical solution

Area of Science:

• * Theoretical Physics
• * Mathematics
• * Network Science

Background:

• * Random-matrix theory (RMT) is crucial for understanding complex systems.
• * Sparse non-Hermitian random matrices model various physical phenomena.
• * Existing RMT results, like the Kesten-McKay measure and Girko's elliptic law, lack sparse generalizations.

Purpose of the Study:

• * To derive the exact analytical expression for the spectrum of a sparse non-Hermitian random matrix ensemble.
• * To generalize the Kesten-McKay measure and Girko's elliptic law to sparse matrices.
• * To investigate the impact of graph symmetry on transport processes in sparse networks.

Main Methods:

• * Exact analytical derivation of the spectral density for sparse non-Hermitian random matrices.
• * Generalization of established RMT results (Kesten-McKay, Girko's elliptic law).
• * Analytical study of transport processes on sparse random graphs with varying edge symmetry.

Main Results:

• * An exact analytical expression for the spectrum of sparse non-Hermitian random matrices was obtained.
• * This expression unifies and extends the Kesten-McKay measure and Girko's elliptic law.
• * The convergence rate of transport processes on sparse graphs exhibits non-monotonic dependence on edge symmetry.

Conclusions:

• * The derived spectral formula provides a powerful tool for analyzing sparse non-Hermitian systems.
• * The study offers new insights into physical problems modeled by sparse random graphs.
• * Non-monotonic dependence of transport rates on graph symmetry highlights complex network behaviors.