A path integral approach to sparse non-Hermitian random matrices

The theory of large random matrices has proved an invaluable tool for the study of systems with disordered interactions in many quite disparate research areas. Widely applicable results, such as the celebrated elliptic law for dense random matrices, allow one to deduce the statistical properties of the interactions in a complex dynamical system that permit stability. However, such simple and universal results have so far proved difficult to come by in the case of sparse random matrices. Here, we present a new approach, which maps the hermitized resolvent of a random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling one to construct Feynman diagrams and perform a perturbative analysis. This approach provides simple closed-form expressions for the eigenvalue spectrum, allowing one to derive modified versions of the classic elliptic and semi-circle laws that take into account the sparse correction. Additionally, in order to demonstrate the broad utility of the path integral framework, we derive a non-Hermitian generalization of the Marchenko-Pastur law, and we also show how one can handle non-negligible higher-order statistics (i.e. non-Gaussian statistics) in dense ensembles.