Eigenvalue Fluctuations of Random Matrices beyond the Gaussian Universality Class

Eigenvalue Fluctuations of Random Matrices beyond the Gaussian Universality Class Elliot Paquette Chair of the Supervisory Committee: Associate Professor Ioana Dumitriu Mathematics The goal of this thesis is to develop one of the threads of what is known in random matrix theory as universality, which essentially is that a large class of matrices generalizing the Gaussian matrices (certain Wigner matrices and β-ensembles) show the same limiting behavior as the Gaussian ensembles. The more well known thread is the universality of local eigenvalue statistics, which is most directly tied to the physical roots of the theory. The other thread is the universality of the fluctuations of global eigenvalue statistics, which occurs in the same classes of matrices that show the local universality. We focus on this second type of universality, and we analyze in detail some features of the limiting Gaussian process that arises this way. Beyond that, the mathematical contributions of this thesis are in three different models of random matrices, twice visiting linear statistics and once visiting the correct order scaling of the spectrum. In all cases, the ensembles studied do not fall into a matrix class covered by current, broad universality theorems. The closest of these models to the Gaussian class is the β-Jacobi ensemble. It lies firmly in the environment of classical random matrix theory in that it can be defined by a log-gas with potential V (x) = −(p − 1) log(x) − (q − 1) log(1−x) for some p and q. From the standpoint of Johansson’s results, the interest here is to see if the singular constraining potential is strong enough to disrupt the global fluctuations. We see that this is not the case, and the same formula for CLTs of linear statistics holds. We additionally find the limiting level density, and we find the first order correction to the limiting level density, which is also obtained by Johansson. The next of these models is the adjacency matrix of a permutation model regular graph. This matrix can be defined by sampling independently and uniformly permutation matrices P1, P2, . , Pd and defining Pn,d := P1 + P t 1 + P2 + P t 2 + · · ·+ Pd + P t d. For this matrix we will show how to derive a uniform bound on the eigenvalues that holds for all n and d and show how this bound can be used in conjunction with estimates on the number of non-backtracking walks to derive the law of the global fluctuations. In this setting, the dependence of d on n is seen to govern whether or not the d-regular graph has Gaussian-type global fluctuations. If d → ∞ slowly as n → ∞, then the fluctuations are like the GOE. On the other hand, if d remains fixed, the fluctuations are a Poissonian analogue. Finally, we will investigate the normalized Laplacian L of the Erdős–Rényi graph model, a graph on n vertices with edges included independently and with probability p = p(n). The behavior of this object depends strongly on whether or not the degrees are strongly concentrated around their means. Indeed, the most interesting features of this graph from the spectral point of view arise precisely when the degrees stop being strongly concentrated. When all the degrees are concentrated, (the p = Ω(log n/n) regime), then the nontrivial eigenvalues of L are within a 1/√np window of 1, consistent with GOE predictions. Outside of this regime, the answer is less clear. If p is exactly log n/n, the graph has isolated vertices with probability tending to 1/e. If the graph has isolated vertices, then L has eigenvalue 0 with multiplicity greater than 1, violating GOE predictions. This thesis will estimate the nontrivial eigenvalues of L as p descends to log n/n and attempt to determine their order. Each of these problems has additional motivations outside of the scope of exploring the Gaussian universality class. Each problem will be presented with its own history and motivation in addition to its contribution to understanding the broader picture.