Related papers: Random Normal Matrices and Polynomial Curves
Let $G$ be the interior domain of a piecewise analytic Jordan curve without cusps. Let $\{p_n\}_{n=0}^\infty$ be the sequence of polynomials that are orthonormal over $G$ with respect to the area measure, with each $p_n$ having leading…
We prove that if a positive closed current is bounded by another one with bounded, continuous or Hoelder continuous super-potentials, then it inherits the same property. There are two different methods to define wedge-products of positive…
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…
It is well known that certain pairs of planar domains have the same spectra of the Laplacian operator. We prove that these domains are still isospectral for a wider class of physical problems, including the cases of heterogeneous drums and…
In this paper, we derive nearly tight probabilistic norm bounds for a class of random matrices we call graph matrices. While the classical case of symmetric matrices with independent random entries (Wigner's matrices) is a special case, in…
In July 2012, at the Conference on Applications of Graph Spectra in Computer Science, Barcelona, D. Stevanovic posed the following open problem: which graphs have the zero as the largest eigenvalue of their modularity matrix? The conjecture…
It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing…
We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the…
Our aim is to study matrix polynomials over max-algebras and their growth in terms of a max-induced semi-norm. We investigate the relationship between the asymptotic growth of polynomial products and the joint spectral radius of the…
We study a class of holomorphic matrix models. The integrals are taken over middle dimensional cycles in the space of complex square matrices. As the size of the matrices tends to infinity, the distribution of eigenvalues is given by a…
It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the…
This paper is concerned with the asymptotic behavior of the free energy for a class of Hermitean random matrix models, with odd degree polynomial potential, in the large N limit. It continues an investigation initiated and developed in a…
Random matrix models consisting of normal matrices, defined by the sole constraint $[N^{\dag},N]=0$, will be explored. It is shown that cubic eigenvalue repulsion in the complex plane is universal with respect to the probability…
For a large class of symmetric random matrices with correlated entries, selected from stationary random fields of centered and square integrable variables, we show that the limiting distribution of eigenvalue counting measure always exists…
Eigenvalues of stochastic matrices have been studied from two complementary perspectives. The individual eigenvalues are characterised through the well-established Karpelevich regions. The spectrum as a whole has also been analysed,…
We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only…
In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a "polynomial ellipsoid" (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only…
We show that with high probability the number of real zeroes of a random polynomial is bounded by the number of vertices on its Newton-Hadamard polygon times the cube of the logarithm of the polynomial degree. A similar estimate holds for…
We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint. We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix…
Let $D$ be a domain with fraction field $K$, and let $M_n(D)$ be the ring of $n \times n$ matrices with entries in $D$. The ring of integer-valued polynomials on the matrix ring $M_n(D)$, denoted ${\rm Int}_K(M_n(D))$, consists of those…