Explicit expressions are proven for higher-order and mixed derivatives of determinant and Pfaffian ratios over Vandermonde determinants in random matrix theory.
Replica treatment of non-Hermitian disordered Hamiltonians
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abstract
We employ the fermionic and bosonic replicated nonlinear sigma models to treat Ginibre unitary, symplectic, and orthogonal ensembles of non-Hermitian random matrix Hamiltonians. Using saddle point approach combined with Borel resummation procedure we derive the exact large-N results for microscopic density of states in all three ensembles. We also obtain tails of the density of states as well the two-point function for the unitary ensemble.
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Derivative relations for determinants, Pfaffians and characteristic polynomials in random matrix theory
Explicit expressions are proven for higher-order and mixed derivatives of determinant and Pfaffian ratios over Vandermonde determinants in random matrix theory.