相关论文: Ratios of characteristic polynomials in complex ma…

Using a character expansion method, we calculate exactly the eigenvalue density of random matrices of the form M^\dagger M where M is a complex matrix drawn from a normalized distribution P(M) ~ exp(-\Tr(A M B M^\dagger) with A and B…

Thom (residual) polynomials in characteristic classes are used in the analysis of geometry of functional spaces. They serve as a tool in description of classes Poincar\'e dual to subvarieties of functions of prescribed types. We give…

The paper is concerned with the correlation functions of the characteristic polynomials of random matrices with independent complex entries. We investigate how the asymptotic behavior of the correlation functions depends on the second…

We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial…

It has been shown recently [10] that Cauchy transforms of orthogonal polynomials appear naturally in general correlation functions containing ratios of characteristic polynomials of random NxN Hermitian matrices. Our main goal is to…

We utilize Cauchy's argument principle in combination with the Jacobian of a holomorphic function in several complex variables and the first moment of a ratio of two correlated complex normal random variables to prove explicit formulas for…

We study the characteristic polynomial of Haar distributed random unitary matrices. We show that after a suitable normalization, as one increases the size of the matrix, powers of the absolute value of the characteristic polynomial as well…

The eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalised to discrete settings involving either a linear or exponential lattice. The corresponding correlation functions can be expressed…

A new family of asymmetric matrices of Walsh-Hadamard type is introduced. We study their properties and, in particular, compute their determinants and discuss their eigenvalues. The invertibility of these matrices implies that certain…

We consider the set $\mathcal{M}_n(\mathbb{Z}; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper and lower bounds on the number of distinct irreducible characteristic polynomials which correspond to…

Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to…

We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…

It is proved that the roots of combinations of matrix polynomials with real roots can be recast as eigenvalues of combinations of real symmetric matrices, under certain hypotheses. The proof is based on recent solution of the Lax…

We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…

We obtain several Cauchy-like and Pellet-like results for the zeros of a general complex polynomial by considering similarity transformations of the squared companion matrix and the reformulation of the zeros of a scalar polynomial as the…

We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that…

When a randomness is introduced at the level of real matrix elements, depending on its particular realization, a pair of eigenvalues can appear as real or form a complex conjugate pair. We show that in the limit of large matrix size the…

We introduce the concept of reflexive moment functional in two variables and the definition of reflexive orthogonal polynomial system. Also reverse matrices and their interesting algebraic properties are studied. Reverse matrices and…

A recursive method is derived to calculate all eigenvalue correlation functions of a random hermitian matrix in the large size limit, and after smoothing of the short scale oscillations. The property that the two-point function is…

The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…