Discrete Orthogonal Polynomials: Asymptotics and Applications (Annals of Mathematics Studies, 164) - Softcover

Baik, J.; Kriecherbauer, T.; McLaughlin, Kenneth Dean T-R; Miller, Peter D.

 
9780691127347: Discrete Orthogonal Polynomials: Asymptotics and Applications (Annals of Mathematics Studies, 164)
Softcover
ISBN 10: 0691127344 ISBN 13: 9780691127347
Publisher: Princeton University Press, 2007

Synopsis

This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case.

J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.

"synopsis" may belong to another edition of this title.

About the Author

J. Baik is Associate Professor of Mathematics at the University of Michigan. T. Kriecherbauer is Professor of Mathematics at Ruhr-Universität Bochum in Bochum, Germany. K. T.-R. McLaughlin is Professor of Mathematics at the University of Arizona. P. D. Miller is Associate Professor of Mathematics at the University of Michigan.

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Discrete Orthogonal Polynomials

Asymptotics and Applications

By J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin, P. D. Miller

PRINCETON UNIVERSITY PRESS

Copyright © 2007 Princeton University Press
All rights reserved.
ISBN: 978-0-691-12734-7

Contents

Preface....................................................................vii
Chapter 1. Introduction....................................................1
Chapter 2. Asymptotics of General Discrete Orthogonal Polynomials in the 
Complex Plane..............................................................		25
Chapter 3. Applications....................................................49
Chapter 4. An Equivalent Riemann-Hilbert Problem...........................67
Chapter 5. Asymptotic Analysis.............................................87
Chapter 6. Discrete Orthogonal Polynomials: Proofs of Theorems Stated in 
§2.3.......................................................................		105
Chapter 7. Universality: Proofs of Theorems Stated in §3.3.................115
Appendix A. The Explicit Solution of Riemann-Hilbert Problem 5.1...........135
Appendix B. Construction of the Hahn Equilibrium Measure: the Proof of 
Theorem 2.17...............................................................		145
Appendix C. List of Important Symbols......................................153
Bibliography...............................................................163
Index......................................................................167

Excerpt

<h2>CHAPTER 1</h2><p>Introduction</p><br><p><b>1.1 MOTIVATING APPLICATIONS</b></p><p>The main aim of this monograph is to deduce asymptotic properties of polynomials that are orthogonal withrespect to pure point measures supported on finite sets and use them to establish various statistical propertiesof discrete orthogonal polynomial ensembles, a special case of which yields new results for a random rhombustiling of a large hexagon. Throughout this monograph, the polynomials that are orthogonal with respect topure point measures will be referred to simply as <i>discrete orthogonal polynomials</i>. We begin by introducingseveral applications in which asymptotics of discrete orthogonal polynomials play an important role.</p><br><p><b>1.1.1 Discrete orthogonal polynomial ensembles</b></p><p>In order to illustrate some concrete applications of discrete orthogonal polynomials and also to provide somemotivation for the scalings we study in this book, we give here a brief introduction to discrete orthogonalpolynomial ensembles. More details can be found in Chapter 3.</p><br><p><i>General theory</i></p><p>In the theory of random matrices [Meh91, Dei99], the main object of study is the joint probability distributionof the eigenvalues. In unitary-invariant matrix ensembles, the eigenvalues are distributed as a Coulomb gasin the plane confined on the real line at the inverse temperature β = 2 subject to an external field. Inrecent years, various problems in probability theory have turned out to be representable in terms of thesame Coulomb gas system with the condition that the particles are further confined to a discrete set. Sucha system is called a <i>discrete orthogonal polynomial ensemble</i>. More precisely, consider the joint probabilitydistribution of finding <i>k</i> particles at positions <i>x<sub>1</sub>, ..., x<sub>k</sub></i> in a discrete set <i>X</i> to be given by the followingexpression (we are using the symbol P(event) to denote the probability of an event):</p><p><i>p<sup>(k)</sup>(x<sub>1</sub>, ..., x<sub>k</sub>)</i> := P (there are particles at each of the nodes <i>x<sub>1</sub>, ..., x<sub>k</sub></i>)</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1)</p><p>where <i>Z<sub>k</sub></i> is a normalization constant (or <i>partition function</i>) chosen so that</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><p>Note that the particles are all indistinguishable from each other.</p><p>Discrete orthogonal polynomial ensembles arise in a number of specific contexts (see, for example, [BorO01,Joh00, Joh01, Joh02]), with particular choices of the weight function <i>w</i>(·) related (in cases we are aware of)to classical discrete orthogonal polynomials. For instance:</p><p>• The Meixner weight</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><p>arises in the directed last-passage site percolation model in the two-dimensional finite lattice Z<sub><i>M</i></sub> x Z<sub><i>N</i></sub>with independent geometric random variables as passage times for each site [Joh00]. The rightmostnode occupied by a particle in the ensemble, <i>x</i><sub>max</sub> := max<i><sub>j</sub>x<sub>j</sub></i> , is a random variable having the samedistribution as the last passage time to travel from the site (0, 0) to the site <i>(M - 1, N - 1)</i>.</p><p>• The Charlier weight</p><p><i>w(x) = t<sup>x</sup>/x!, for x = 0, 1, 2, ...,</i></p><p>arises in the longest random word problem [Joh01].</p><p>• The Krawtchouk weight</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],</p><p>arises in the random domino tiling of the Aztec diamond [Joh01, Joh02].</p><p>• The Hahn weight</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><p>arises in the random rhombus tiling of a hexagon [Joh01, Joh02]. See also §3.4 for more details.</p><br><p>The first two cases (Meixner and Charlier) are examples of the <i>Schur measure</i> [BorO01, Oko01] on the setof partitions. On the other hand, in special limiting cases the Meixner and Charlier ensembles both becomethe <i>Plancherel measure</i>, which describes the longest increasing subsequence of a random permutationn[BaiDJ99, BorOO00, Joh01]. Clearly it would be of some theoretical interest to determine properties of theensembles that are more or less independent of the particular choice of weight function, at least within someclass. Such properties are said to support the conjecture of <i>universality</i> within the class of weight functionsunder consideration.</p><p>As iss well known in random matrix theory [Meh91, TraW98], many quantities such as correlation functionsand gap probabilities are expressible in terms of the polynomials that are orthogonal with respect to theweight w. In a similar way, the asymptotic study of discrete orthogonal polynomial ensembles is translatedto the asymptotic study of discrete orthogonal polynomials. In many applications,,,, the parameters of theweight are varying, and at the same time the location of the particle of interest also varies. For example,in the Hahn case, the interesting case turns out to be when <i>N</i> -> ∞ and the location of a particle isscaled as <i>x = aN</i> + [xi] or <i>x = aN</i> + [xi]<i>N</i><sup>1/3</sup> for some constant <i>a</i> > 0. Equivalently, upon scaling by1/<i>N</i>, the discrete set <i>X/N</i> asymptotically fills out an interval, while <i>x/N</i> scales as <i>x/N = a</i> + [xi]/<i>N</i> or<i>x/N = a</i> + [xi]<i>N</i><sup>-2/3</sup>. Under this scaling limit (Plancherel-Rotach asymptotics), the asymptotics of Meixner,Charlier, and Krawtchouk polynomials were obtained using integral representations [IsmS98, Joh00, Joh02].However, even though the Hahn polynomials are also classical polynomials, their integral representation doesnot seem to be so straightforward to analyze asymptotically using the classical steepest-descent method,and the asymptotics have not yet been obtained. The subject of this book is a general class of weights thatcontains Krawtchouk and Hahn weights (and a lot more) as special cases. For these general weights, we willcompute the asymptotics of the associated discrete orthogonal polynomials for all values of the variable zin the complex plane. By specializing to suitable scaling regimes, the desired asymptotics of the associateddiscrete orthogonal polynomial ensembles are obtained.</p><br><p><i>Random tilings</i></p><p>The discrete orthogonal polynomial ensemble with the Hahn weight represents a probability distributionfor certain events in a random rhombus tiling of a hexagon, and new results on the asymptotics of randomtilings will be presented in Chapter 3 using this connection. Here, we briefly introduce the subject of randomrhombus tilings of a hexagon.</p><p>Let a, b, and c be positive integers and consider the hexagon illustrated in Figure 1.1 having the followingvertices (written as points in the complex plane):</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><p>All interior angles of this hexagon are equal and measure 2π/3 radian, and the lengths of the sides are,starting with the side <i>(P<sub>1</sub>, P<sub>2</sub>)</i> and proceeding in counterclockwise order, b, a, c, b, a, c. We call this theabc-<i>hexagon</i>. Denote by L the part of the set of lattice points</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><p>that lies within the hexagon, including the sides <i>(P<sub>6</sub>, P<sub>1</sub>), (P<sub>1</sub>, P<sub>2</sub>),(P<sub>2</sub>, P<sub>3</sub>)</i>, and <i>(P<sub>3</sub>, P<sub>4</sub>)</i> but excluding thesides <i>(P<sub>4</sub>, P<sub>5</sub>)</i> and <i>(P<sub>5</sub>, P<sub>6</sub>)</i>. The lattice L can also be seen in Figure 1.1.</p><p>Consider covering the abc-hexagon with rhombus tiles having sides of unit length. The rhombus tilescome in three different types (orientations), which we refer to as type I, type II, and type III as shown inFigure 1.2. Tiles of types I and II are sometimes collectively called <i>horizontal rhombi</i>, while tiles of type IIIare sometimes called vertical rhombi. The position of each rhombus tile in the hexagon is a specific latticepoint in L defined as indicated in Figure 1.2.</p><p>MacMahon's formula [Mac60] gives the total number of all possible rhombus tilings of the abc-hexagon asthe expression</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><p>Consider the set of all rhombus tilings equipped with uniform probability. Hence we choose a tiling of theabc-hexagon at random. It is of some current interest to determine the behavior of various correspondingstatistics of this ensemble in the limit as a, b, c -> ∞.</p><p>In the scaling limit of <i>n</i> -> ∞, where</p><p><i>a = An, b = Bn , c = Cn,</i> (1.2)</p><p>with fixed A,B, C > 0, the regions near the six corners are <i>frozen</i> or <i>polar zones</i> (i.e., regions in whichonly one type of tile is present), while toward the center of the hexagon is a <i>temperate zone</i> (i.e., a regioncontaining all three types of tiles). The random tiling shown in Figure 1.3 dramatically illustrates the twotypes of regions, and the asymptotically sharp nature of the boundary between them, the <i>Arctic circle</i>.Indeed, it was shown by Cohn, Larsen, and Propp [CohLP98] that in such a limit, upon scaling by 1/<i>n</i>, theexpected shape of the boundary separating the polar zones from the temperate zone is given by the inscribedellipse. The next interesting problem would be to compute the limiting fluctuation of the boundary. Whilethis problem had remained unsolved until our work, an analogous problem had been solved in the context ofrectangle tilings of Aztec diamonds; it is proved in [Joh01] that the fluctuation of the boundary between thepolar zones and the temperate zone in the Aztec diamond tiling model is governed (in a proper scaling limit)by the Tracy-Widom law in random matrix theory [TraW94]. Indeed, Johansson [Joh01, Joh02] expressedthe induced probability for certain configurations of rectangles in an Aztec diamond and of rhombi in theabc-hexagon in terms of particular discrete orthogonal polynomial ensembles. The weights correspondingto the Aztec diamond are Krawtchouk weights, and those corresponding to the abc-hexagon are Hahn orassociated Hahn weights. By applying the classical steepest-descent method to the integral representation ofthe Krawtchouk polynomials, Johansson obtained the Tracy-Widomdistribution for the Aztec diamond. Oneof the results implied by our analysis of general discrete orthogonal polynomial ensembles (see Theorem 3.14)is that the same Tracy-Widom law holds for rhombus tilings of the abc-hexagon. More details of theconnection between the statistics of rhombus tilings and the Hahn discrete orthogonal polynomial ensemblesare discussed along with our asymptotic results in §3.4 and, specifically, §3.4.2.</p><br><p><b>1.1.2 The continuum limit of the Toda lattice</b></p><p>In Flaschka's variables, the Toda lattice equations are the following coupled nonlinear ordinary differentialequations:</p><p><i>d<sub>ak</sub>/dt = 2b<sup>2</sup><sub>k</sub> - 2b<sup>2</sup><sub>k-1</sub></i> (1.3)</p><p>and</p><p><i>db<sub>k</sub>/dt = (a<sub>k+1</sub> - a<sub>k</sub>)b<sub>k</sub>.</i> (1.4)</p><p>Here <i>k</i> is an integer index. The finite Toda lattices arise by "cutting the chain", with the imposition ofboundary conditions at, say, <i>k</i> = 0 and <i>k = N</i> - 1. For one type of boundary condition, we may assumethat with <i>a<sub>k</sub> = a<sub>N,k</sub></i> and <i>b<sub>k</sub> = b<sub>N,k</sub></i>, (1.4) holds for<i>k</i> = 0, ..., <i>N</i> - 2 and (1.3) holds for <i>k</i> = 1, ..., <i>N</i> - 2,while</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)</p><p>Thus we obtain a first-order system of nonlinear differential equationsfor unknowns <i>a<sub>N,0</sub>(t), ..., a<sub>N,N-1</sub>(t)</i>and <i>b<sub>N,0</sub>(t), ..., b<sub>N,N-2</sub>(t).</i></p><p>One way to view the Toda lattice equations (1.3) and (1.4) is as a numerical scheme for integrating thehyperbolic system</p><p>[partial derivative]A/[partial derivative]T = 4B [partial derivative]B/[partial derivative]cand [partial derivative]B/[partial derivative]T = B [partial derivative]A/[partial derivative]c. (1.6)</p><p>Here <i>T = t/N</i> is a rescaled time, 1/<i>N</i> is a small lattice spacing, and <i>c = k/N</i>. Of course, as (1.6) is anonlinear hyperbolic system of partial differential equations, initially smooth solutions of (1.6) can develop<i>shocks</i> (derivative singularities) in finite time. When this occurs, the Toda lattice equations should no longerbe viewed as a viable numerical method for studying weak solutions of (1.6), as rapid oscillations develop inthe numerical solution that are (it turns out) inconsistent even in an averaged sense with viscosity solutionsof the hyperbolic system. See Figures 1.4 and 1.5.</p><p>The proof of convergence of the Toda numerical scheme for (1.6), for times <i>T</i> less than the shock time,is one of the results of the analysis carried out by Deift and McLaughlin [DeiM98]. Their analysis of theToda lattice equations with smooth initial data goes beyond the shock time and characterizes precisely theaverage properties of the rapid oscillations that subsequently occur. Indeed, it turns out that, while rapidlyvarying (on the fast time scale <i>t</i> and on the grid scale <i>k</i>), these oscillations can nonetheless be characterizedby slowly varying quantities that satisfy an enlarged set of hyperbolic nonlinear partial differential equations(the Whitham equations) generalizing the hyperbolic system (1.6). The continuum limit of the Toda latticehas also been considered from a geometric point of view [BloGPU03] and from the point of view of orthogonalpolynomials [AptV01].</p><p>The method used in [DeiM98] exploits the fact that the Toda lattice equations (1.3) and (1.4) comprisea completely integrable system. Specifically, this fact implies closed-form formulae for <i>a<sub>N,k</sub>(t)</i> and <i>b<sub>N,k</sub>(t)</i>in terms of initial data via ratios of Hankel-type determinants. Deift and McLaughlin analyzed thesedeterminantal formulae in the continuum limit <i>N</i> -> ∞ for initial data sampling fixed smooth functions<i>A(c)</i> and <i>B(c)</i> > 0 given on the interval <i>c</i> [member of] [0, 1]. Using the Lax-Levermore method, they showed thatthe large-<i>N</i> asymptotics are characterized by the solution of a constrained variational problem for a certainextremal measure on an interval, which they solved by converting it into a scalar Riemann-Hilbert problem.The number of subintervals where the extremal measure is unconstrained (this number depends generally on<i>c</i> and <i>T</i>) turns out to be related to the size of the system of Whitham equations needed to describe the slowlyvarying features of the microscopic oscillations. In particular, when there is only one such subinterval, thelimit <i>N</i> -> ∞ is strong, and the hyperbolic system (1.6) governs the limit. Shock formation corresponds tothe splitting of one unconstrained subinterval into two. Once this occurs, the analysis in [DeiM98] establishesthe continuum limit only in a weak (averaged) sense. One of the applications that we will describe in thismonograph is the strengthening of the asymptotics obtained in [DeiM98] after the shock time; we will obtainstrong (locally uniform) asymptotics for the aN,k(t) and bN,k(t) even in the oscillatory region appearing aftera shock has occurred, for example, in the irregular regions of the final plots in Figures 1.4 and 1.5.</p><p>Dynamical stability of solutions of the Toda lattice equations may be studied by means of the linearizedToda equations. Substituting <i>a<sub>N,k</sub>(t) -> a<sub>N,k</sub>(t) + a<sub>N,k</sub>(t)</i>and <i>b<sub>N,k</sub>(t) -> b<sub>N,k</sub>(t)(1 + b<sub>N,k</sub>(t))</i> into (1.3) and(1.4) and keeping only the linear terms gives</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.7)</p><p>and</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.8)</p><p>Linear stability analysis is interesting in the large-N limit both before and after shock formation. Unfortunately,the detailed analysis in [DeiM98] does not directly provide information about a complete basisfor solutions of the linearized equations (1.7) and (1.8) corresponding to given smooth functions <i>A(c)</i> and<i>B(c)</i>. One of the consequences of the analysis described in this book is an asymptotic description of thesolutions of the linearized Toda equations in the continuum limit both before and after shock time. In someapplications the linearized problem can have meaning as a linear system in its own right with prescribedtime-dependent potentials <i>a<sub>N,k</sub>(t)</i> and <i>b<sub>N,k</sub>(t)</i>; it is mathematically convenient then to imagine that thepotentials originate from a Toda solution, in which case there is a complete basis for time-dependent modes(<i>i.e.</i>, a time-dependent spectral transform) for the linear problem. This approach is described for other completelyintegrable systems in [MilA98] and [MilC01], with corresponding physical applications described inreferences therein. See §3.5 for details of the results implied by application of the analysis in this monographto the continuum limit of the Toda lattice.</p><p><i>(Continues...)</i>

(Continues...)
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Princeton University Press
Publication date
2007
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English
ISBN 10 
0691127344
ISBN 13 
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Number of pages
184

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