The Ginibre ensemble and Gaussian analytic functions

We show that as $n$ changes, the characteristic polynomial of the $n\times n$ random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to P\'olya's urn scheme. As a result, we get a random analytic function in the limit, which is given by a mixture of Gaussian analytic functions. This gives another reason why the zeros of Gaussian analytic functions and the Ginibre ensemble exhibit similar local repulsion, but different global behavior. Our approach gives new explicit formulas for the limiting analytic function.Comment: 23 pages, 1 figur

Finite term relations for the exponential orthogonal polynomials

The exponential orthogonal polynomials encode via the theory of hyponormal operators a shade function $g$ supported by a bounded planar shape. We prove under natural regularity assumptions that these complex polynomials satisfy a three term relation if and only if the underlying shape is an ellipse carrying uniform black on white. More generally, we show that a finite term relation among these orthogonal polynomials holds if and only if the first row in the associated Hessenberg matrix has finite support. This rigidity phenomenon is in sharp contrast with the theory of classical complex orthogonal polynomials. On function theory side, we offer an effective way based on the Cauchy transforms of $g, \overline{z}g, \ldots, \overline{z}^dg$, to decide whether a $(d+2)$-term relation among the exponential orthogonal polynomials exists; in that case we indicate how the shade function $g$ can be reconstructed from a resulting polynomial of degree $d$ and the Cauchy transform of $g$. A discussion of the relevance of the main concepts in Hele-Shaw dynamics completes the article.Comment: 33 page

Circular Jacobi Ensembles and deformed Verblunsky coefficients

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: c_{\delta,\beta}^{(n)} \prod_{1\leq k with $\Re \delta > -1/2$. If $e$ is a cyclic vector for a unitary $n\times n$ matrix $U$, the spectral measure of the pair $(U,e)$ is well parameterized by its Verblunsky coefficients $(\alpha_0, ..., \alpha_{n-1})$. We introduce here a deformation $(\gamma_0, >..., \gamma_{n-1})$ of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product $r(\gamma_0)... r(\gamma_{n-1})$ of elementary reflections parameterized by these coefficients. If $\gamma_0, ..., \gamma_{n-1}$ are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above. These deformed Verblunsky coefficients also allow to prove that, in the regime $\delta = \delta(n)$ with \delta(n)/n \to \dd, the spectral measure and the empirical spectral distribution weakly converge to an explicit nontrivial probability measure supported by an arc of the unit circle. We also prove the large deviations for the empirical spectral distribution.Comment: New section on large deviations for the empirical spectral distribution, Corrected value for the limiting free energ

Upper Hessenberg and Toeplitz Bohemians

We also acknowledge the support of the Ontario Graduate Institution, The National Science & Engineering Research Council of Canada, the University of Alcala, the Rotman Institute of Philosophy, the Ontario Research Centre of Computer Algebra, and Western University. Part of this work was developed while R. M. Corless was visiting the University of Alcala, in the frame of the project Giner de los Rios. J.R. Sendra is member of the Research Group ASYNACS (Ref. CT-CE2019/683).A set of matrices with entries from a fixed finite population P is called “Bohemian”. The mnemonic comes from BOunded HEight Matrix of Integers, BOHEMI, and although the population P need not be solely made up of integers, it frequently is. In this paper we look at Bohemians, specifically those with population {−1,0,+1} and sometimes other populations, for instance {0,1,i,−1,−i}. More, we specialize the matrices to be upper Hessenberg Bohemian. We then study the characteristic polynomials of these matrices, and their height, that is the infinity norm of the vector of monomial basis coefficients. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the number of normal matrices and the number of stable matrices in these families.Agencia Estatal de Investigació

The matrix equation XA − BX = R and its applications

AbstractWe study the well-known Sylvester equation XA − BX = R in the case when A and B are given and R is known up to its first n - 1 rows. We prove new results on the existence and uniqueness of X. Our results essentially state that, in case A is a nonderogatory matrix, there always exists a solution to this equation; a solution is uniquely determined by its first row x1; and there is an interesting relationship between x1 and the rows of R. We also give a complete characterization of the nonsingularity of X in this case. As applications of our results we develop direct methods for constructing symmetrizers and commuting matrices, computing the characteristic polynomial of a matrix, and finding the numbers of common eigenvalues between A and B. Some well-known important results on symmetrizers, Bezoutians, and inertia are recovered as special cases

Matrix models for circular ensembles

We describe an ensemble of (sparse) random matrices whose eigenvalues follow the Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature beta. Our approach combines elements from the theory of orthogonal polynomials on the unit circle with ideas from recent work of Dumitriu and Edelman. In particular, we resolve a question left open by them: find a tri-diagonal model for the Jacobi ensemble.Comment: 28 page