Complex Eigenvalues of High Dimensional Quaternion Random Matrices
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This thesis is concerned with complex eigenvalues of quaternion random matrices in the limit of large matrix dimensions. Three random matrix ensembles are considered: quaternion Ginibre, induced quaternion Ginibre and truncated unitary symplectic matrices. For these ensembles, eigenvalue density and correlation functions are obtained in the bulk of eigenvalue distribution and, in some cases, at its boundaries. Eigenvalue statistics of quaternionic matrices have no rotational invariance, which makes their analysis difficult. Almost all of the results available in the literature are for radial eigenvalue statistics. The main novelty of this thesis is that the asymptotic analysis is performed without averaging over the angle which gives access to the eigenvalue depletion area near the real axis. The thesis has five chapters. Chapter 1 offers an introduction to random matrices and quaternions. Chapter 2 recalls known relevant results for matrices with complex entries. Chapter 3 revisits the quaternion Ginibre ensemble and introduces the main ideas and tools for eigenvalue analysis. These tools are then tested to obtain the eigenvalue density at the boundary of the eigenvalue distribution (semi-circular arc, real line and the corner points where the arc meets the real line). Chapter 4 is concerned with the induced quaternion Ginibre ensemble. It is proved that in the limit of large matrix dimensions the eigenvalues are uniformly distributed inside an annulus; and the eigenvalue correlation functions in the bulk and also in the eigenvalue depletion area coincide with those for the symplectic Ginibre ensemble. Chapter 5 is concerned with truncations of unitary symplectic matrices: eigenvalue jpdf is derived in closed form for finite matrix dimensions, and the eigenvalue density and correlation functions are obtained in the limit of large matrix dimension. It turns out that the eigenvalue correlation functions in the bulk in this ensemble coincide with those in the complex Ginibre ensemble.
Authors
Lysychkin, SCollections
- Theses [4190]