Riemann-Hilbert Analysis Research Articles

Multiple scale asymptotics of map enumeration

We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two asymptotic expansions obtained from two different fields of mathematics: the Riemann–Hilbert analysis of orthogonal polynomials and the theory of discrete dynamical systems. By equating the coefficients of these expansions in a common region of uniform validity in their parameters, we recover known results and provide new expressions for generating functions associated with graphical enumeration on surfaces of genera 0 through 7. Although the body of the article focuses on 4-valent maps, the methodology presented here extends to regular maps of arbitrary even valence and to some cases of odd valence, as detailed in the appendices.

A Periodic Hexagon Tiling Model and Non-Hermitian Orthogonal Polynomials

We study a one-parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling. The description of the asymptotic behavior can be separated into two regimes: the low and the high temperature regime. Our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there. For low temperatures, the disordered region consists of two disjoint ellipses. In the high temperature regime the two ellipses merge into a single simply connected region. At the transition from the low to the high temperature a tacnode appears. The key to our asymptotic study is a recent approach introduced by Duits and Kuijlaars providing a double integral representation for the correlation kernel. One of the factors in the integrand is the Christoffel–Darboux kernel associated to polynomials that satisfy non-Hermitian orthogonality relations with respect to a complex-valued weight on a contour in the complex plane. We compute the asymptotic behavior of these orthogonal polynomials and the Christoffel–Darboux kernel by means of a Riemann–Hilbert analysis. After substituting the resulting asymptotic formulas into the double integral we prove our main results by classical steepest descent arguments.

Asymptotics of orthogonal polynomials with asymptotic Freud‐like weights

AbstractWe derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann–Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.

The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials

AbstractIn this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight urn:x-wiley:00222526:media:sapm12259:sapm12259-math-0001The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near , the uniform asymptotic expansion involves Airy function as , and Bessel function of order α as in the neighborhood of , the uniform asymptotic expansion is associated with Bessel function of order β as . The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painlevé III transcendent. We also obtain the limit of the kernel in the bulk of the spectrum. The double scaled logarithmic derivative of the Hankel determinant satisfies a σ‐form Painlevé III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method.

Riemann-Hilbert analysis for a Nikishin system

In this paper we give the asymptotic behaviour of type I multiple orthogonal polynomials for a Nikishin system of order two with two disjoint intervals. We use the Riemann-Hilbert problem for multiple orthogonal polynomials and steepest descent analysis for oscillatory Riemann- Hilbert problems to obtain the asymptotic behaviour in all the relevant regions of the complex plane. Bibliography: 38 titles.

Nonintersecting Brownian bridges on the unit circle with drift

Nonintersecting Brownian bridges on the unit circle form a determinantal stochastic process exhibiting random matrix statistics for large numbers of walkers. We investigate the effect of adding a drift term μ to walkers on the circle conditioned to start and end at the same position. For each return time T<π2 we show there exists a critical drift μc(T) such that if |μ−2πm/T|<μc(T) for some integer m then the expected winding number for each walker is asymptotically m. In addition, we compute the asymptotic distribution of total winding numbers in the double-scaling regime in which the expected number of walkers with winding number not equal to m is finite. The method of proof is Riemann–Hilbert analysis of a certain family of discrete orthogonal polynomials with varying complex exponential weights. This is the first asymptotic analysis of such a class of polynomials. We determine asymptotic formulas for these polynomials as the degree of the polynomial grows large and demonstrate the emergence of a second band of zeros by a mechanism not previously seen for discrete orthogonal polynomials with real weights.

Multiple orthogonal polynomials on the unit circle. Normality and recurrence relations

Multiple orthogonal polynomials on the unit circle (MOPUC) were introduced by J. Mínguez and W. Van Assche for the first time in 2008. Some applications were given there and recurrence relations were obtained from a Riemann–Hilbert problem.This paper is a second contribution to this field. We first obtain a determinantal formula for MOPUC (multiple Heine’s formula) and we analyze the concept of normality, from a dynamical point of view and by presenting a first example: the combination of the Lebesgue and Rogers–Szegő measures. Secondly, we deduce recurrence relations for MOPUC without using Riemann–Hilbert analysis, only by considering orthogonality conditions. This new approach allows us to complete the recurrence relations in the situation when the origin is a zero of MOPUC, a case that was not considered before. As a consequence, we give an appropriate definition of multiple Verblunsky coefficients. A multiple version of the well known Szegő recurrence relation is also obtained. Here, the coefficients that appear in the recurrence satisfy certain partial difference equations that are used to present a recursive algorithm for the computation of MOPUC. A discussion on the Riemann–Hilbert approach that also includes the case when the origin is a zero of MOPUC is presented. Some conclusions and open questions are finally mentioned.

Classical Liouville three-point functions from Riemann-Hilbert analysis

We study semiclassical correlation functions in Liouville field theory on a two-sphere when all operators have large conformal dimensions. In the usual approach, such computation involves solving the classical Liouville equation, which is known to be extremely difficult for higher-point functions. To overcome this difficulty, we propose a new method based on the Riemann-Hilbert analysis, which is applied recently to the holographic calculation of correlation functions in AdS/CFT. The method allows us to directly compute the correlation functions without solving the Liouville equation explicitly. To demonstrate its utility, we apply it to three-point functions, which are known to be solvable, and confirm that it correctly reproduces the classical limit of the DOZZ formula for quantum three-point functions. This provides good evidence for the validity of this method.

Nonintersecting Brownian Motions on the Half-Line and Discrete Gaussian Orthogonal Polynomials

We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the Airy2 process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.

Orthogonal trigonometric polynomials: Riemann–Hilbert analysis and relations with OPUC

Orthogonal trigonometric polynomials: Riemann–Hilbert analysis and relations with OPUC

Universality in complex Wishart ensembles for general covariance matrices with 2 distinct eigenvalues

We considered N × N Wishart ensembles in the class W C ( Σ N , M ) (complex Wishart matrices with M degrees of freedom and covariance matrix Σ N ) such that N 0 eigenvalues of Σ N are 1 and N 1 = N − N 0 of them are a . We studied the limit as M , N , N 0 and N 1 all go to infinity such that N M → c , N 1 N → β and 0 < c , β < 1 . In this case, the limiting eigenvalue density can either be supported on 1 or 2 disjoint intervals in R + , and a phase transition occurs when the support changes from 1 interval to 2 intervals. By using the Riemann–Hilbert analysis, we have shown that when the phase transition occurs, the eigenvalue distribution is described by the Pearcey kernel near the critical point where the support splits.

First Colonization of a Hard-Edge in Random Matrix Theory

We describe the spectral statistics of the first finite number of eigenvalues in a newly-forming band on the hard-edge of the spectrum of a random Hermitean matrix model, a phenomenon also known as the “birth of a cut” near a hard-edge. It is found that in a suitable scaling regime, they are described by the same spectral statistics of a finite-size Laguerre-type matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials.

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t* the smallest paths hit the hard edge and from then on are stuck to it. For t ≠ t* we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights

The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject. In the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a Riemann–Hilbert analysis. In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees.

First Colonization of a Spectral Outpost in Random Matrix Theory

We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N −2γ where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature. The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity at the rate N −γ , whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.

Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour

Classical Jacobi polynomials Pn(α, β), with α,β > - 1, have a number of well-known properties, in particular the location of their zeros in the open interval (-1, 1). This property is no longer val...

Riemann–Hilbert analysis for Jacobi polynomials orthogonal on a single contour

Classical Jacobi polynomials P n ( α , β ) , with α , β > - 1 , have a number of well-known properties, in particular the location of their zeros in the open interval ( - 1 , 1 ) . This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters α n , β n depend on n in such a way that lim n → ∞ α n n = A , lim n → ∞ β n n = B , with A , B ∈ R . We restrict our attention to the case where the limits A , B are not both positive and take values outside of the triangle bounded by the straight lines A = 0 , B = 0 and A + B + 2 = 0 . As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential. The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift–Zhou steepest descent method based on the Riemann–Hilbert reformulation of Jacobi polynomials.

Riemann–Hilbert analysis and uniform convergence of rational interpolants to the exponential function

We study the asymptotic behavior of the polynomials p and q of degrees n, rational interpolants to the exponential function, defined by p ( z ) e - z / 2 + q ( z ) e z / 2 = O ( ω 2 n + 1 ( z ) ) , as z tends to the roots of ω 2 n + 1 , a complex polynomial of degree 2 n + 1 . The roots of ω 2 n + 1 may grow to infinity with n, but their modulus should remain uniformly bounded by c log ( n ) , c < 1 / 2 , as n → ∞ . We follow an approach similar to the one in a recent work with Arno Kuijlaars and Walter Van Assche on Hermite–Padé approximants to e z . The polynomials p and q are characterized by a Riemann–Hilbert problem for a 2 × 2 matrix valued function. The Deift–Zhou steepest descent method for Riemann–Hilbert problems is used to obtain strong uniform asymptotics for the scaled polynomials p ( 2 nz ) and q ( 2 nz ) in every domain in the complex plane. From these asymptotics, we deduce uniform convergence of general rational interpolants to the exponential function and a precise estimate on the error function. This extends previous results on rational interpolants to the exponential function known so far for real interpolation points and some cases of complex conjugate interpolation points.

On convergence of moments for random Young tableaux and a random growth model

In recent work of Baik, Deift and Rains convergence of moments was established for the limiting joint distribution of the lengths of the first k rows in random Young tableaux. The main difficulty was obtaining a good estimate for the tail of the distribution and this was accomplished through a highly nontrival Riemann-Hilbert analysis. Here we give a simpler derivation. The same method is used to establish convergence of moments for a random growth model.

The asymptotics of monotone subsequences of involutions

We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions are, depending on the number of fixed points, (1) the Tracy-Widom distributions for the largest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy-Widom distributions. We also consider the second rows of the corresponding Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of J. Baik and E. Rains in [7] which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the Riemann-Hilbert analysis for the orthogonal polynomials by P. Deift, K. Johansson, and Baik in [3].