Spectral properties of a non-Hermitian extension of the diluted Wishart ensemble

Abstract We develop a theoretical framework based on the cavity and replica methods to analyze the spectral properties of sparse asymmetric correlation matrices of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="bold-italic">F</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">X</mml:mi> <mml:msup> <mml:mi mathvariant="bold-italic">Y</mml:mi> <mml:mi mathvariant="normal">⊤</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>ω</mml:mi> <mml:mi mathvariant="bold-italic">Y</mml:mi> <mml:msup> <mml:mi mathvariant="bold-italic">X</mml:mi> <mml:mi mathvariant="normal">⊤</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> , where X and Y are adjacency matrices of weighted Erdős–Rényi random graphs. We examine how the spectral density evolves as the asymmetry parameter ω varies from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> (nearly symmetric matrices) to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>ω</mml:mi> <mml:mtext>⩽</mml:mtext> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> (nearly antisymmetric matrices). Analytical predictions are validated through exact numerical diagonalization, showing excellent agreement with theoretical results in the thermodynamic limit.