Logarithmic Potential Theory Research Articles

Dynamics of polynomial semigroups: measures, potentials, and external fields

In this paper, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh-Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Along the way, we establish the continuity of the logarithmic potential for the Dinh-Sibony measure, which might be of independent interest. We then use the F-functional of Mhaskar and Saff to discuss bounds on the capacity and diameter of the Julia sets of such semigroups.

A generalization of the discrete Rodrigues formula for Meixner polynomials

A generalization of Meixner polynomials leading to a new construction of Apéry approximations is put forward. The limiting distribution of the zeros of scaled polynomials is described in terms of algebraic functions. The resulting distribution is shown to be a solution of some vector equilibrium problem in the theory of logarithmic potential. Bibliography: 21 titles.

Intersection of unit balls in classical matrix ensembles

We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and Schmuckenschlager for classical lp-balls [Schechtman and Schmuckenschlager, GAFA Lecture Notes, 1991]. The proof of our result is based on two ingredients, which are of independent interest. The first one is a weak law of large numbers for a point chosen uniformly at random in the unit ball of such a matrix ensemble. The second one is an explicit computation of the asymptotic volume of such matrix unit balls, which in turn is based on the theory of logarithmic potentials with external fields.

Exact asymptotic volume and volume ratio of Schatten unit balls

The unit ball Bpn(R) of the finite-dimensional Schatten trace class Spn consists of all real n×n matrices A whose singular values s1(A),…,sn(A) satisfy s1p(A)+…+snp(A)≤1, where p>0. Saint Raymond (1984) showed that the limit limn→∞n1∕2+1∕p(VolBpn(R))1∕n2 exists in (0,∞) and provided both lower and upper bounds. In this manuscript we use the theory of logarithmic potentials in external fields to determine the precise limiting constant and thus the exact asymptotic volume of Bpn(R). The corresponding result for complex Schatten balls is also obtained. As an application we compute the precise asymptotic volume ratio of the Schatten p-balls, as n→∞, thereby extending Saint Raymond’s estimate in the case of the nuclear norm (p=1) to the full regime 1≤p≤∞ with exact limiting behavior.

Zero Distribution of Orthogonal Polynomials on a q-Lattice

We give the asymptotic behavior of the zeros of orthogonal polynomials, after appropriate scaling, for which the orthogonality measure is supported on the $q$-lattice $\{q^k, k=0,1,2,3,\ldots\}$, where $0 < q < 1$. The asymptotic distribution of the zeros is given by the radial part of the equilibrium measure of an extremal problem in logarithmic potential theory for circular symmetric measures with a constraint imposed by the $q$-lattice.

Potentials on a Compact Riemann Surface

Fundamental concepts of potential theory on compact Riemann surfaces are defined that generalize the corresponding concepts of logarithmic potential theory on the complex plane. The standard properties of these quantities are proved, and relationships between them are established.

Minkowski’s question mark measure

Minkowski’s question mark function is the distribution function of a singular continuous measure: we study this measure from the point of view of logarithmic potential theory and orthogonal polynomials. We conjecture that it is regular, in the sense of Ullman–Saff–Stahl–Totik and moreover that it belongs to a Nevai class; we provide numerical evidence of the validity of these conjectures. In addition, we study the zeros of its orthogonal polynomials and the associated Christoffel functions, for which asymptotic formulae are derived. As a by-product, we compute upper and lower bounds to the Hausdorff dimension of Minkowski’s measure. Rigorous results and numerical techniques are based upon Iterated Function Systems composed of Möbius maps.

Asymptotics of multipoint Hermite–Padé approximants of the first type for two beta functions

The asymptotic behavior of the Hermite–Pade approximants of the first type for two beta functions are studied. The results are expressed in terms of equilibrium problems of logarithmic potential theory and in terms of meromorphic functions on Riemann surfaces.

Modified Logarithmic Potential Theory and Applications

We develop potential theory including a Bernstein-Walsh type estimate for functions of the form $p(z)q(f(z))$ where $p,q$ are polynomials and $f$ is holomorphic. Such functions arise in the study of certain ensembles of probability measures and our estimates lead to probabilistic results such as large deviation principles.

Orthogonal polynomials of equilibrium measures supported on Cantor sets

The equilibrium measure of a compact set is a fundamental object in logarithmic potential theory. We compute numerically this measure and its orthogonal polynomials, when the compact set is a Cantor set, defined by an Iterated Function System.We first construct sequences of discrete measures, via the solution of large systems of non-linear equations, that converge weakly to the equilibrium measure. Successively, we compute their Jacobi matrices via standard procedures, enhanced for the scope. Numerical estimates of the convergence rate to the limit Jacobi matrix are provided, which show stability and efficiency of the whole procedure. As a companion result, we also compute Jacobi matrices in two other cases: equilibrium measures on finite sets of intervals, and balanced measures of Iterated Function Systems.These algorithms can reach large polynomial orders: therefore, we study the asymptotic behavior of the orthogonal polynomials and, by a natural extension of the concept of regular root asymptotics, we derive an efficient scheme for the computation of complex Green’s functions and of related conformal mappings.

Logarithmic Potential Theory and Large Deviation

We derive a general large deviation principle for a canonical sequence of probability measures, having its origins in random matrix theory, on unbounded sets $K$ of ${\bf C}$ with weakly admissible external fields $Q$ and very general measures $\nu$ on $K$. For this we use logarithmic potential theory in ${\bf R}^{n}$, $n\geq 2$, and a standard contraction principle in large deviation theory which we apply from the two-dimensional sphere in ${\bf R}^{3}$ to the complex plane ${\bf C}$.

Families of vector measures which are equilibrium measures in an external field

We consider vector extremal problems in the theory of logarithmic potential with external field by looking at an example of two-dimensional problems with Nikishin interaction matrix and variable masses and of the first and second components of the vector measure, respectively. The dependence of the supports of the equilibrium measures, equlibrium constants and energy on the parameter is analysed. Integral formulae recovering the extremal measure with mass from the supports of extremal measures with smaller masses are obtained. Bibliography: 27 titles.

Random normal matrices and Ward identities

Consider the random normal matrix ensemble associated with a potential on the plane which is sufficiently strong near infinity. It is known that, to a first approximation, the eigenvalues obey a certain equilibrium distribution, given by Frostman's solution to the minimum energy problem of weighted logarithmic potential theory. On a finer scale, one can consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of fluctuations, and we prove that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.

Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval

We consider polynomials pnω(x) that are orthogonal with respect to the oscillatory weight w(x)=eiωx on [−1,1], where ω>0 is a real parameter. A first analysis of pnω(x) for large values of ω was carried out in Asheim et al. (2014), in connection with complex Gaussian quadrature rules with uniform good properties in ω. In this contribution we study the existence, asymptotic behavior and asymptotic distribution of the roots of pnω(x) in the complex plane as n→∞. The parameter ω grows with n linearly. The tools used are logarithmic potential theory and the S-property, together with the Riemann–Hilbert formulation and the Deift–Zhou steepest descent method.

Apéry Polynomials and the Multivariate Saddle Point Method

The Apéry polynomials and in particular their asymptotic behavior play an essential role in the understanding of the irrationality of $$\zeta (3)$$ . In this paper, we present a method to study the asymptotic behavior of the sequence of Apéry polynomials $$(B_{n})_{n=1}^{\infty }$$ in the whole complex plane as $$n\rightarrow \infty $$ . The proofs are based on a multivariate version of the complex saddle point method. Moreover, the asymptotic zero distributions for the polynomials $$(B_{n})_{n=1}^{\infty }$$ and for some transformed Apéry polynomials are derived by means of the theory of logarithmic potentials with external fields, establishing a characterization as the unique solution of a weighted equilibrium problem. The method applied is a general one, so that the treatment can serve as a model for the study of objects related to the Apéry polynomials.

S-curves in polynomial external fields

Curves in the complex plane that satisfy the S-property were first introduced by Stahl and they were further studied by Gonchar and Rakhmanov in the 1980s. Rakhmanov recently showed the existence of curves with the S-property in a harmonic external field by means of a max–min variational problem in logarithmic potential theory. This is done in a fairly general setting, which however does not include the important special case of an external field ReV where V is a polynomial of degree ≥2. In this paper we give a detailed proof of the existence of a curve with the S-property in the external field ReV within the collection of all curves that connect two or more pre-assigned directions at infinity in which ReV→+∞. Our method of proof is very much based on the works of Rakhmanov on the max–min variational problem and of Martínez-Finkelshtein and Rakhmanov on critical measures.

LARGE DEVIATIONS FOR A NON-CENTERED WISHART MATRIX

We investigate an additive perturbation of a complex Wishart random matrix and prove that a large deviation principle holds for the spectral measures. The rate function is associated to a vector equilibrium problem coming from logarithmic potential theory, which in our case is a quadratic map involving the logarithmic energies, or Voiculescu's entropies, of two measures in the presence of an external field and an upper constraint. The proof is based on a two type particles Coulomb gas representation for the eigenvalue distribution, which gives a new insight on why such variational problems should describe the limiting spectral distribution. This representation is available because of a Nikishin structure satisfied by the weights of the multiple orthogonal polynomials hidden in the background.

Weakly admissible vector equilibrium problems

We establish lower semi-continuity and strict convexity of the energy functionals for a large class of vector equilibrium problems in logarithmic potential theory. This, in particular, implies the existence and uniqueness of a minimizer for such vector equilibrium problems. Our work extends earlier results in that we allow unbounded supports without having strongly confining external fields. To deal with the possible noncompactness of supports, we map the vector equilibrium problem onto the Riemann sphere and our results follow from a study of vector equilibrium problems on compacts in higher dimensions. Our results cover a number of cases that were recently considered in random matrix theory and for which the existence of a minimizer was not clearly established yet.

RANDOM RIGHT EIGENVALUES OF GAUSSIAN QUATERNIONIC MATRICES

We consider a random matrix whose entries are independent Gaussian variables taking values in the field of quaternions with variance 1/n. Using logarithmic potential theory, we prove the almost sure convergence, as the dimension n goes to infinity, of the empirical distribution of the right eigenvalues towards some measure supported on the unit ball of the quaternions field. Some comments on more general Gaussian quaternionic random matrix models are also made.

Weakly admissible vector equilibrium problems

We establish lower semi-continuity and strict convexity of the energy functionals for a large class of vector equilibrium problems in logarithmic potential theory. This, in particular, implies the existence and uniqueness of a minimizer for such vector equilibrium problems. Our work extends earlier results in that we allow unbounded supports without having strongly confining external fields. To deal with the possible noncompactness of supports, we map the vector equilibrium problem onto the Riemann sphere and our results follow from a study of vector equilibrium problems on compacts in higher dimensions. Our results cover a number of cases that were recently considered in random matrix theory and for which the existence of a minimizer was not clearly established yet.