arXiv subjects

Maxim L. Yattselev

Publications and source records attributed to Maxim L. Yattselev.

29 records · Page 2Linked to original sources

arXiv2018

The reciprocal Mahler ensembles of random polynomials

We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree $N$ whose Mahler measure is bounded by a constant. After a change of variables this reduces to a generalization of Ginibre's complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval $[-2,2]$ on the real axis in the complex plane. In the complex (real) case the random roots form a determinantal (Pfaffian) point process, and in both cases the empirical measure on roots converges weakly to the arcsine distribution supported on $[-2,2]$. Outside this region the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from $[-2,2]$. These kernels, as well as the scaling limits for the kernels in the bulk $(-2,2)$ and at the endpoints $\{-2,2\}$ are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels.

math.CA↗

arXiv2017

Zeros of real random polynomials spanned by OPUC

Let $ \{φ_i\}_{i=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure $ μ$. We study zero distribution of random linear combinations of the form \[ P_n(z)=\sum_{i=0}^{n-1}η_iφ_i(z), \] where $ η_0,\dots,η_{n-1} $ are i.i.d. standard Gaussian variables. We use the Christoffel-Darboux formula to simplify the density functions provided by Vanderbei for the expected number real and complex of zeros of $ P_n $. From these expressions, under the assumption that $ μ$ is in the Nevai class, we deduce the limiting value of these density functions away from the unit circle. Under the mere assumption that $ μ$ is doubling on subarcs of $ \T $ centered at $ 1 $ and $ -1 $, we show that the expected number of real zeros of $ P_n $ is at most \[ (2/π) \log n +O(1), \] and that the asymptotic equality holds when the corresponding recurrence coefficients decay no slower than $ n^{-(3+ε)/2} $, $ ε>0 $. We conclude with providing results that estimate the expected number of complex zeros of $ P_n $ in shrinking neighborhoods of compact subsets of $ \T $.

math.CA↗

arXiv2017

Symmetric Contours and Convergent Interpolation

The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as applied to the multipoint Padé approximants is the fact that given a germ of an algebraic function and a sequence of rational interpolants with free poles of the germ, if there exists a contour that is "symmetric" with respect to the interpolation scheme, does not separate the plane, and in the complement of which the germ has a single-valued continuation with non-identically zero jump across the contour, then the interpolants converge to that continuation in logarithmic capacity in the complement of the contour. The existence of such a contour is not guaranteed. In this work we do construct a class of pairs interpolation scheme/symmetric contour with the help of hyperelliptic Riemann surfaces (following the ideas of Nuttall \& Singh and Baratchart \& the author. We consider rational interpolants with free poles of Cauchy transforms of non-vanishing complex densities on such contours under mild smoothness assumptions on the density. We utilize $ \bar\partial $-extension of the Riemann-Hilbert technique to obtain formulae of strong asymptotics for the error of interpolation.

math.CA↗

arXiv2016

Szegő-type asymptotics for ray sequences of Frobenius-Padé approximants

Let $\widehatσ$ be a Cauchy transform of a possibly complex-valued Borel measure $σ$ and $\{p_n\}$ be a system of orthonormal polynomials with respect to a measure $μ$, $\mathrm{supp}(μ)\cap\mathrm{supp}(σ)=\varnothing$. An $(m,n)$-th Frobenius-Padé approximant to $\widehatσ$ is a rational function $P/Q$, $\mathrm{deg}(P)\leq m$, $\mathrm{deg}(Q)\leq n$, such that the first $m+n+1$ Fourier coefficients of the linear form $Q\widehatσ-P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Padé approximants to $\widehatσ$ along ray sequences $\frac n{n+m+1}\to c>0$, $n-1\leq m$, when $μ$ and $σ$ are supported on intervals on the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the respective interval are holomorphic functions.

math.CA↗

arXiv2015

Strong Asymptotics of Hermite-Padé Approximants for Angelesco Systems

In this work type II Hermite-Padé approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi-indices.

math.CA↗

arXiv2015

Hermite-Padé approximants for a pair of Cauchy transforms with overlapping symmetric supports

Hermite-Padé approximants of type II are vectors of rational functions with common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szegő-type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann-Hilbert problem for multiple orthogonal polynomials (the common denominator).

math.CA↗

arXiv2014

Nuttall's theorem with analytic weights on algebraic S-contours

Given a function $f$ holomorphic at infinity, the $n$-th diagonal Padé approximant to $f$, denoted by $[n/n]_f$, is a rational function of type $(n,n)$ that has the highest order of contact with $f$ at infinity. Nuttall's theorem provides an asymptotic formula for the error of approximation $f-[n/n]_f$ in the case where $f$ is the Cauchy integral of a smooth density with respect to the arcsine distribution on [-1,1]. In this note, Nuttall's theorem is extended to Cauchy integrals of analytic densities on the so-called algebraic S-contours (in the sense of Nuttall and Stahl).

math.CA↗

arXiv2013

Root Statistics of Random Polynomials with Bounded Mahler Measure

The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree $N$ polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yields a Pfaffian point process on the complex plane. When $N$ is large, with probability tending to 1, the roots tend to the unit circle, and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle. When this point is away from the real axis (on which there is a positive probability of finding a root) the scaled process degenerates to a determinantal point process with the same local statistics (i.e., scalar kernel) as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of Mahler measure at most 1. Three new matrix kernels appear in a neighborhood of $\pm 1$ which encode information about the correlations between real roots, between complex roots and between real and complex roots. Away from the unit circle, the kernels converge to new limiting kernels, which imply among other things that the expected number of roots in any open subset of $\C$ disjoint from the unit circle converges to a positive number. We also give ensembles with identical statistics drawn from two-dimensional electrostatics with potential theoretic weights, and normal matrices chosen with regard to their topological entropy as actions on Euclidean space.

math-ph↗

arXiv2012

Pade approximants for functions with branch points - strong asymptotics of Nuttall-Stahl polynomials

Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, f \in\mathcal{A}(\bar{\C} \setminus A), \sharp A <\infty. J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single-valued branch and the domain of convergence of the diagonal Pade approximants for f. The Pade approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of f in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function f single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has minimal logarithmic capacity among all other systems converting the function f to a single-valued branch. Thus the domain of convergence corresponds to the maximal (in the sense of minimal boundary) domain of single-valued holomorphy for the analytic function f \in\mathcal{A}(\bar{\C} \setminus A). The complete proof of Nuttall's conjecture (even in a more general setting where the set A has logarithmic capacity zero) was obtained by H. Stahl. In this work, we derive strong asymptotics for the denominators of the diagonal Pade approximants for this problem in a rather general setting. We assume that A is a finite set of branch points of f which have the algebraic character and which are placed in a generic position. The last restriction means that we exclude from our consideration some degenerated "constellations" of the branch points.

math.CA↗

arXiv2012

Large Deviations and Linear Statistics for Potential Theoretic Ensembles Associated with Regular Closed Sets

A two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged regular closed region K whose charge density is determined by its equilibrium potential at an inverse temperature βis investigated. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a point process in the complex plane. We describe the weak* limits of the joint intensities of this point process and show that it is exponentially likely to find the process in a neighborhood of the equilibrium measure for K.

math.CA↗

arXiv2012

Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary

We investigate a two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region K whose charge density is determined by its equilibrium potential at an inverse temperature corresponding to β= 2. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a determinantal point process on the complex plane. We investigate the scaling limit, as N \to \infty, of the associated kernel in the neighborhood of a point on the boundary under the assumption that the boundary is sufficiently smooth. We find that the limiting kernel depends on the limiting value of N/s, and prove universality for these kernels. That is, we show that, the scaled kernel in a neighborhood of a point ζ\in \partial K can be succinctly expressed in terms of the scaled kernel for the closed unit disk, and the exterior conformal map which carries the complement K to the complement of the closed unit disk. When N / s \to 0 we recover the universal kernel discovered by Doron Lubinsky in Universality type limits for Bergman orthogonal polynomials, Comput. Methods Funct. Theory, 10:135-154, 2010.

math.CA↗