Sparse non-Hermitian random matrices

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The goal of this thesis is to study and explore several open problems in the statistical analysis of non-Hermitian random matrices. The most studied example of non-Hermitian random matrices is the Ginibre ensemble and its generalization to the real and symplectic case. However there are rich classes of non-Hermitian random matrices of significant interest.Such matrices occur in at least two distinct settings: they describe the spectra of non-equilibrium systems that interact through a complex network structure; they correspond to the Lax matrices of integrable systems with random initial data.In the thesis the spectrum of sparse non-Hermitian random matrices and its asymptotic properties will be studied.The work map is to compute the spectrum of significant classes of non-Hermitian random matrices, obtain numerical insghts, formulate new conjectures and prove rigorous results.