Asymptotic Distribution Of Zeros Research Articles

Applications of multiple orthogonal polynomials with hypergeometric moment generating functions

We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval, the positive real line, or the unit circle and the multiple orthogonal polynomials are generalizations of the Jacobi, Laguerre or Bessel orthogonal polynomials. We give explicit formulas for the type I and type II multiple orthogonal polynomials and study some of their properties. In particular, we describe the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials via the free convolution. Essential to our overall approach is the use of the Mellin transform. Finally, we discuss two applications. First, we show that the multiple orthogonal polynomials appear naturally in the study of the squared singular values of (mixed) products of truncated unitary random matrices and Ginibre matrices. Secondly, we use the multiple orthogonal polynomials to simultaneously approximate certain hypergeometric series and to provide an explicit proof of their Q-linear independence.

Winding Number, Density of States, and Acceleration

Abstract Winding number and density of states are two fundamental physical quantities for non-self-adjoint quasi-periodic Schrödinger operators, which reflect the asymptotic distribution of zeros of the characteristic determinants of the truncated operators under Dirichlet boundary condition, with respect to complexified phase and the energy, respectively. We will prove that the winding number is in fact Avila’s acceleration, it is also closely related to the density of states by a generalized Thouless formula for non-self-adjoint Schrödinger operators and Avila’s global theory. Communicating Editor: Prof Svetlana Jitomirskaya

Mod-φ convergence of Stirling distributions and limit theorems for zeros of their generating functions

We study mod-φ convergence of several probability distributions on the set of positive integers that involve Stirling numbers of both kinds and, as a consequence, derive various limit theorems for these distributions. We also derive closely related limit theorems for the distribution of zeros of the corresponding generating functions. For example, we identify the asymptotic distribution of zeros for the generating polynomial of the number of occupied boxes when n balls are allocated equiprobably and independently among θ boxes in the regime when θ grows linearly with n.

Multiple orthogonal polynomials associated with the exponential integral

AbstractWe introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights on the positive real line, with the gamma density and a density related to the exponential integral . We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed‐type multiple orthogonal polynomials.

Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds

Given a sequence of positive Hermitian holomorphic line bundles (Lp,hp) on a Kähler manifold X, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of Lp, under a natural convergence assumption on the sequence of curvatures c1(Lp,hp). We then apply this to study the asymptotic distribution of common zeros of random sequences of m-tuples of sections of Lp as p→+∞.

Polynomials of Least Deviation from Zero in Sobolev p-Norm

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p-norm (1<p<infty ) for the case p=1. Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that the n-th polynomial of least deviation has at least n-mathbf {d}^* zeros on the convex hull of the support of the measure, where mathbf {d}^* denotes the number of terms in the discrete part.

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

AbstractA new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval (0,1) is studied. This type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total‐positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson‐type differential equation. The focus is on the polynomials whose indices lie on the step‐line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. We obtain Rodrigues‐type formulas for type I polynomials and functions, while a more detailed characterization is given for the type II polynomials (aka 2‐orthogonal polynomials) that include an explicit expression as a terminating hypergeometric series, a third‐order differential equation, and a third‐order recurrence relation. The asymptotic behavior of their recurrence coefficients mimics those of Jacobi–Piñeiro polynomials, based on which their asymptotic zero distribution and a Mehler–Heine asymptotic formula near the origin are given. Particular choices of the parameters and confluence relations give some known systems such as special cases of the Jacobi–Piñeiro polynomials, Jacobi‐type 2‐orthogonal polynomials, components of the cubic decomposition of threefold symmetric Hahn‐classical polynomials, and multiple orthogonal polynomials with respect to confluent hypergeometric functions of the second kind.

The asymptotic zero distribution of Lommel polynomials as functions of their order with a variable complex argument

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence of the weak limit of corresponding root-counting measures and deduce formulas for the supporting curves and density. The obtained result represents a solvable example of a more general problem which is still open. Numerical illustrations of the main result are also involved.

Asymptotic zero distribution for a class of extremal polynomials

We consider extremal polynomials with respect to a Sobolev-type [Formula: see text]-norm, with [Formula: see text] and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures (i.e. supported on disjoint subsets of the real line), it is proved that their critical points are simple and contained in the interior of the convex hull of the support of the measures involved and the asymptotic critical point distribution is studied. We also find the [Formula: see text]th root asymptotic behavior of the corresponding sequence of Sobolev extremal polynomials and their derivatives.

Supercritical regime for the kissing polynomials

We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function eniλz on [−1,1], where λ is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when λ is smaller than a certain critical value, λc. Our main goal is to compute their asymptotics when λ>λc.We first provide a geometric description, based on the theory of quadratic differentials, of the curves in the complex plane which will eventually support the asymptotic zero distribution of these polynomials. Next, using the powerful Riemann–Hilbert formulation of the orthogonal polynomials due to Fokas, Its, and Kitaev, along with its method of asymptotic solution via Deift–Zhou nonlinear steepest descent, we provide uniform asymptotics of the polynomials throughout the complex plane.Although much of this asymptotic analysis follows along the lines of previous works in the literature, the main obstacle appears in the construction of the so-called global parametrix. This construction is carried out in an explicit way with the help of certain integrals of elliptic type. In stark contrast to the situation one typically encounters in the presence of real orthogonality, an interesting byproduct of this construction is that there is a discrete set of values of λ for which one cannot solve the model Riemann–Hilbert problem, and as such the corresponding polynomials fail to exist.

Universality results for zeros of random holomorphic sections

In this work we prove a universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact Kähler complex space X X . Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of the probability measure on the space of holomorphic sections. In the case when X X is a compact Kähler manifold, we also prove an off-diagonal exponential decay estimate for the Bergman kernels of a sequence of positive line bundles on X X .

Mass Equidistribution for Random Polynomials

The purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of associated orthogonal polynomials with subgaussian coefficients. This class of probability distributions contains a wide range of random variables including standard Gaussian and all bounded random variables. We prove that for almost every sequence of random polynomials their normalized zero currents become equidistributed with respect to a deterministic extremal current. The main ingredients of the proof are Bergman kernel asymptotics, mass equidistribution of random polynomials and concentration inequalities for subgaussian quadratic forms.

Laguerre–Angelesco multiple orthogonal polynomials on an [formula omitted]-star

We investigate the type I and type II multiple orthogonal polynomials on an r-star with weight function |x|βe−xr, with β>−1. Each measure μj, for 1≤j≤r, is supported on the semi-infinite interval [0,ωj−1∞) with ω=e2πi∕r. For both the type I and the type II polynomials we give explicit expressions, the coefficients in the recurrence relation, the differential equation and we obtain the asymptotic zero distribution of the polynomials on the diagonal. Also, we give the connection between the Laguerre–Angelesco polynomials and the Jacobi–Angelesco polynomials on an r-star.

Asymptotic zero distribution of random orthogonal polynomials

We consider random polynomials of the form $H_{n}(z)=\sum_{j=0}^{n}\xi_{j}q_{j}(z)$ where the $\{\xi_{j}\}$ are i.i.d. nondegenerate complex random variables, and the $\{q_{j}(z)\}$ are orthonormal polynomials with respect to a compactly supported measure $\tau $ satisfying the Bernstein–Markov property on a regular compact set $K\subset\mathbb{C}$. We show that if $\mathbb{P}(|\xi_{0}|>e^{|z|})=o(|z|^{-1})$, then the normalized counting measure of the zeros of $H_{n}$ converges weakly in probability to the equilibrium measure of $K$. This is the best possible result, in the sense that the roots of $G_{n}(z)=\sum_{j=0}^{n}\xi_{j}z^{j}$ fail to converge in probability to the appropriate equilibrium measure when the above condition on the $\xi_{j}$ is not satisfied. We also consider random polynomials of the form $\sum_{k=0}^{n}\xi_{k}f_{n,k}z^{k}$, where the coefficients $f_{n,k}$ are complex constants satisfying certain conditions, and the random variables $\{\xi_{k}\}$ satisfy $\mathbb{E}\log (1+|\xi_{0}|)<\infty $. In this case, we establish almost sure convergence of the normalized counting measure of the zeros to an appropriate limiting measure. Again, this is the best possible result in the same sense as above.

Critical measures for vector energy: Asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weight

We consider the type I multiple orthogonal polynomials (MOPs) (An,m,Bn,m), deg⁡An,m≤n−1, deg⁡Bn,m≤m−1, and type II MOPs Pn,m, deg⁡Pn,m=n+m, satisfying non-hermitian orthogonality with respect to the weight e−z3 on two unbounded contours γ1 and γ2 on C, with (in the case of type II MOPs) n conditions on γ1 and m on γ2. Under the assumption thatn,m→∞,nn+m→α∈(0,1) we find the detailed (rescaled) asymptotics of An,m, Bn,m and Pn,m on C, and describe the phase transitions of this limit behavior as a function of α. This description is given in terms of the vector critical measure μ→α=(μ1,μ2,μ3), the saddle point of an energy functional comprising both attracting and repelling forces. This critical measure is characterized by a cubic equation (spectral curve), and its components μj live on trajectories of a canonical quadratic differential ϖ on the Riemann surface of this equation. The structure of these trajectories and their deformations as functions of α was object of study in our previous paper [42], and some of the present results strongly rely on the analysis carried out there.We conclude that the asymptotic zero distribution of the polynomials An,m and Pn,m are given by appropriate combinations of the components μj of the vector critical measure μ→α. However, in the case of the zeros of Bn,m the behavior is totally different, and can be described in terms of the balayage of the signed measure μ2−μ3 onto certain curves on the plane. These curves are constructed with the aid of the canonical quadratic differential ϖ, and their topology has three very distinct characters, depending on the value of α, and are obtained from the critical graph of ϖ.Once the trajectories and vector critical measures are studied, the main asymptotic technical tool is the Deift-Zhou nonlinear steepest descent analysis of a 3×3 matrix-valued Riemann–Hilbert problem characterizing both (An,m,Bn,m) and Pn,m, which allows us to obtain the limit behavior of both types of MOPs simultaneously.We illustrate our findings with results of several numerical experiments, explain the computational methodology, and formulate some conjectures and empirical observations based on these experiments.

Multiple Hermite polynomials and simultaneous Gaussian quadrature

Multiple Hermite polynomials are an extension of the classical Hermite polynomials for which orthogonality conditions are imposed with respect to $r>1$ normal (Gaussian) weights $w_j(x)=e^{-x^2+c_jx}$ with different means $c_j/2$, $1 \\leq j \\leq r$. These polynomials have a number of properties such as a Rodrigues formula, recurrence relations (connecting polynomials with nearest neighbor multi-indices), a differential equation, etc. The asymptotic distribution of the (scaled) zeros is investigated, and an interesting new feature is observed: depending on the distance between the $c_j$, $1 \\leq j \\leq r$, the zeros may accumulate on $s$ disjoint intervals, where $1 \\leq s \\leq r$. We will use the zeros of these multiple Hermite polynomials to approximate integrals of the form $\\displaystyle \\int_{-\\infty}^{\\infty} f(x) \\exp(-x^2 + c_jx)\\, dx$ simultaneously for $1 \\leq j \\leq r$ for the case $r=3$ and the situation when the zeros accumulate on three disjoint intervals. We also give some properties of the corresponding quadrature weights.

Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials

We investigate polynomials , , that are multi- orthogonal with respect to a Nikishin system of compactly supported measures over the star-like set of rays . We prove that the Nikishin system is normal, that the polynomials satisfy a three-term recurrence relation of order of the form with for all , and that the nonzero roots of are all simple and located in . Under the assumption that the measures generating the Nikishin system are regular (in the sense of Stahl and Totik), we describe the asymptotic zero distribution and weak behaviour of the polynomials in terms of a vector equilibrium problem for logarithmic potentials. Under the same regularity assumptions, we prove a theorem on the convergence of the Hermite-Padé approximants to the Nikishin system of Cauchy transforms. Bibliography: 16 titles.

On Global universality for zeros of random polynomials

In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations $\sum_{j}a_jP_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j\}_j$ induced by a multi-circular weight function $Q$ satisfying suitable smoothness and growth conditions. In complex dimension $m\geq3$, we prove that $\Bbb{E}[(\log(1+|a_j|))^m]<\infty$ is a necessary and sufficient condition for normalized zero currents of random polynomials to be almost surely asymptotic to the (deterministic) extremal current $dd^cV_{Q}.$ In addition, in complex dimension one, we consider random linear combinations of orthonormal polynomials with respect to a regular measure in the sense of Stahl \& Totik and we prove similar results in this setting.

Distribution of Zeros for Random Laurent Rational Functions

We study the asymptotic distribution of zeros for the random rational functions that can be viewed as partial sums of a random Laurent series. If this series defines a random analytic function in an annulus A, then the zeros accumulate on the boundary circles of A, being equidistributed in the angular sense, with probability 1. We also show that the equidistribution phenomenon holds if the annulus of convergence degenerates to a circle. Moreover, equidistribution of zeros still persists when the Laurent rational functions diverge everywhere, which is new even in the deterministic case. All results hold under two types of general conditions on random coefficients. The first condition is that the random coefficients are non-trivial i.i.d. random variables with finite \(\log ^+\) moments. The second condition allows random variables that need not be independent or identically distributed, but only requires certain uniform bounds on the tails of their distributions.

Zero distribution of random sparse polynomials

We study asymptotic zero distribution of random Laurent polynomials whose support are contained in dilates of a fixed integral polytope $P$ as their degree grow. We consider a large class of probability distributions including the ones induced from i.i.d. random coefficients whose distribution law has bounded density with logarithmically decaying tails as well as moderate measures defined over the projectivized space of Laurent polynomials. We obtain a quantitative localized version of Bernstein-Kouchnirenko Theorem.