In this thesis, we study discrete orthogonal polynomials and their applications. In particular, we investigate asymptotic behaviours of discrete Laguerre polynomials and demonstrate how a model of N non-intersecting Bessel paths is related to discrete and multiple discrete orthogonal polynomials.

In the first part of the thesis, some background and general properties of orthogonal polynomials and non-intersecting Brownian motion are reviewed. By taking n non-intersecting Brownian motions on the half-line as an example, we show that the maximal height is related to discrete Hermite polynomials. Moreover, as the number of particles n goes to infinity, the limiting distribution of the maximal height is the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensembles.

In the second part, we consider $N$ non-intersecting Bessel paths starting at x=a≥0 and conditioned to end at the origin x=0. We derive the explicit formula of the distribution function for the maximum height by applying the Schur function expansion. Depending on the starting point a>0 or a=0, the distribution functions are given in terms of the Hankel determinants associated with the multiple discrete orthogonal polynomials or discrete orthogonal polynomials, respectively.

In the third part, we introduce the Riemann-Hilbert problem and its relation to orthogonal polynomials. The Deift-Zhou nonlinear steepest descent method for the Riemann-Hilbert problem is briefly reviewed, which is the main method we used to derive asymptotics of discrete orthogonal polynomials.

In the fourth part, we study the discrete orthogonal polynomials on the infinite nodes L_n = { x_{k,N} = k²/N², k ∈ Ν} with respect to the Laguerre weight function w(x) = x^α e^{-N c x}, where c is a positive real number. The distribution of the orthogonality nodes is a natural upper constraint for the density of the zeros of discrete orthogonal polynomials. We give asymptotics of the discrete Laguerre polynomials as N →∞ when the upper constraint is attained. Our method is based on the interpolation problem for the discrete orthogonal polynomials, which can be converted into a Riemann-Hilbert problem. We analyze the related Riemann-Hilbert problem by applying the Deift-Zhou nonlinear steepest descent method. Uniform asymptotic expansions are obtained for the discrete Laguerre polynomials in several different regions in the complex plane. The asymptotics for the recurrence coefficients and normalization constant are also derived.

In the last part, we discuss the difference between the continuous orthogonal polynomials and discrete orthogonal polynomials when the upper constraint is not active. For discrete Hermite polynomials, their asymptotic properties agree with the continuous classical Hermite polynomials. However, for discrete Laguerre polynomials, their asymptotic near the origin are different comparing with the continuous Laguerre polynomials. The difference can be seen from some numerical computations of zeros of both polynomials. We also give some insights from the viewpoint of the Riemann-Hilbert analysis, more precisely, the local parametrix construction near the origin.