<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...