Random matrix theory starts from the assumption that the large-scale behaviour of a complex system should be governed by its symmetries and the statistical properties of its parameters, and be relatively insensitive to the precise details of each interacting element. The theory mostly aims to determine the statistics of the eigenvalues and eigenvectors of random matrices in the large-size limit. Early work, originating in nuclear physics, focused on ensembles having both Hermitian symmetry and all-to-all interactions, similar to mean-field models in statistical physics. Relaxing the all-to-all assumption introduces topological disorder and leads to ensembles of sparse random matrices with many zero matrix entries. Such matrices model complex systems where a given degree of freedom interacts with a finite number of others, and arise naturally in connection with systems such as neural networks or ecosystems.
Despite this broad importance, however, sparse non-Hermitian random matrices have only received significant study in the past decade, as standard analysis methods from random matrix theory do not apply. Rigorous results for such matrices are almost non-existent since it is very difficult to prove the convergence of properties of eigenvalues and eigenvectors to a deterministic limit at large matrix sizes. However, recent research has made progress with new approaches. In a new article, LML Fellow Fernando Metz, along with Izaak Neri of King’s College London and Tim Rogers of the University of Bath, review theoretical progress in the study of the spectra of sparse non-Hermitian random matrices, with a focus on exact approaches based on a fruitful analogy between random-matrix calculations and the statistical mechanics of disordered spin systems. As they show, for simple models, these methods give access to analytical results for the spectral properties of sparse non-Hermitian random matrices. For more complicated models, the spectral properties can also be computed in the large-size limit using numerical algorithms.
Metz and colleagues close their review by noting that the theory of sparse non-Hermitian random matrices is still in its infancy, as compared with classical random matrix theory, and there are many outstanding questions. Among them is the matter of universality. Interest in random matrix theory depends largely on the universal behaviour of many spectral observables, which makes it possible to study the stability of complex dynamical systems. In the case of sparse random matrices, this possibility seems to be lost due to strong local fluctuations in the graph structure. However, it turns out that many ensembles of sparse non-Hermitian random matrices do exhibit some universal properties, such as the spectral gap, the eigenvalue with the largest real part, and the eigenvector moments corresponding to this eigenvalue. These spectral properties determine the stability and steady-state dynamics of complex systems. Hence, it appears that there is hope to find universal behaviour for sparse matrices, if one looks at the right observables, which could lead to a better understanding of universality in large dynamical systems.
A pre-print of the paper is available at https://arxiv.org/abs/1811.10416