Let n and N be in N, and for 0 ≤ i ≤ n−1, let αi < βi ∈ R. Consider the monic polynomial in a single complex variable of the form fn(z) = zn + an−1zn−1 +··· +a1z +a0 whose coefficients ai are uniformly distributed on [αiN,βiN]∩Z for each 0 ≤ i ≤ n−1 and jointly independent. This random polynomial model is referred to as the generalized large box model. When instead αij < βij for 1 ≤ i,j ≤ n and one considers the n-by-n random matrix whose entries are uniformly distributed on [αijN,βijN] ∩ Z for each 1 ≤ i,j ≤ n and jointly independent, we say that the matrix is drawn from the generalized large box model ensemble.

This thesis is organized into five chapters. Chapter 1 develops and presents the history of the relevant random polynomial and random matrix models, related results, and notation.

Chapters 2 and 3 are concerned with finding the probability that random polynomials whose coefficients obey the generalized large box model have all real roots, as N → ∞. Specifically, in Chapter 2, discriminant and root analysis methods are applied to low degree polynomials, obtaining explicit answers. These methods further find an extremely dominant root, denoted by ξn, for all degrees; this is a root whose modulus is not tight as N → ∞, while the moduli of the remaining roots are tight as N → ∞. This expands upon on a discovery made by Dubickas and Sha [Exp. Math., 24(3):312–325, 2015]. As N → ∞, we show that ξn is real with probability tending to 1 and that |ξn + an−1| converges in distribution to |X/Y |, where X is uniformly distributed on [αn−2,βn−2], Y is uniformly distributed on [αn−1,βn−1], and X and Y are independent.

In Chapter 3, we consider non-monic degree n−1 polynomials whose coefficients are uniformly distributed on [αi,βi] for 0 ≤ i ≤ n − 1 and jointly independent, referred to as generalized bounded height model polynomials. As N → ∞, the probability that the degree n generalized large box model polynomial with coefficients uniformly distributed on [αiN,βiN] ∩ Z for 0 ≤ i ≤ n−1 and jointly independent has all real roots converges to the probability that the degree n − 1 generalized bounded height model polynomial with coefficients uniformly distributed on [αi,βi] for 0 ≤i≤n−1andjointly independent has all real roots. The methods of Bert´ok, Hajdu, and Peth¨o [J. Number Theory, 179:172–184, 2017] are used to express this probability in terms of an integral formula. For the special case when αi = 0 and βi = 1 for 0 ≤ i ≤ n−1, a relation to the Selberg integral is explored and we show that the probability of such a polynomial having all real roots is positive and monotonically decreasing in n.

In Chapters 4 and 5, the analogous question of the probability that the random matrix whose entries are drawn from the generalized large box model ensemble has all real eigenvalues is considered. In Chapter 4 we begin by letting αij < βij for 1 ≤ i,j ≤ n and consider the n-by n random matrix whose entries are uniformly distributed on [αij,βij] for each 1 ≤ i,j ≤ n and jointly independent. This matrix ensemble is referred to as the generalized bounded height ensemble. Using Edelman’s method [J. Multivariate Anal., 60(2):203–232, 1997], we factor these matrices into their real Schur decomposition and present an integral formula for the probability that generalized bounded height ensemble matrices have all real roots. In Chapter 5, we show that if A is an n-by-n random matrix with entries that are uniformly distributed on [αijN,βijN] ∩ Z for 1 ≤ i,j ≤ n and jointly independent and B is an n-by-n random matrix with entries that are uniformly distributed on [αij,βij] for 1 ≤ i,j ≤ n and jointly independent, then for each 0 ≤ k ≤ n, as N →∞,theprobability that A has exactly k real eigenvalues converges to the probability that B has exactly k real eigenvalues. Moreover, the empirical spectral measure of A/N converges weakly in distribution to the empirical spectral measure of B and the joint distribution of the eigenvalues of A/N converges in distribution to the joint distribution of the eigenvalues of B.

Finally, still in Chapter 5, we consider rank one perturbations of the random matrix A whose entries are all independently and identically (iid) uniformly distributed on [−N,N]∩Z. We say that Ais drawn from the large box model ensemble. Letting P be the matrix whose entries are all µN, we consider the limiting spectral behavior of A+P in the three cases where µN/N → ∞,µN/N → 0, and µN/N → c, for c ∈ R. If lim_N→∞ µN N =∞, the eigenvalues of A+P µN converge almost surely to the eigenvalues of P/µN, and the largest eigenvalue (in magnitude) is real with probability tending to 1. Additionally, the centered and correctly normalized largest eigenvalue of A + P converges in distribution to a Bates distribution with n2 parameters. If lim_N→∞ µN N =0, the empirical spectral measures of A+P N and A N both converge weakly in distribution to the empirical spectral measure of the matrix whose entries are iid and uniformly distributed on [−1,1]; the joint distribution of the eigenvalues of A N and A+P N both converge in distribution to the joint distribution of eigenvalues of the same random matrix. If lim_N→∞ µN N =c>0,the empirical spectral measure of A+P N converges weakly in distribution to the empirical spectral measure of the matrix whose entries are iid and uniformly distributed on [−1 + c,1 + c]. In addition, if c > 12, we show that as N → ∞, A+P N has exactly one real outlier eigenvalue and provide a range for its location.