I'm interested in the behavior of the largest eigenvalue of random $n \times n$ Hermitian matrices of the form $A^{n} = \frac{1}{2}(Q^{n}+(Q^{n})^{\dagger})$ where the real and imaginary components of the entries $Q_{ij}$ are independent and identically distributed random variables drawn from a distribution with support on a bounded interval.
This class of random matrices is closely related to Wigner matrices. The precise definition of Wigner matrices seems to vary from paper to paper, so here I'll quote Tao's lecture notes:
A Wigner Hermitian matrix ensemble is a random matrix ensemble $M_{n} = (\xi_{ij})_{1\leq i,j \leq n}$ of Hermitian matrices ([...]), in which the upper-triangular entries $\xi_{ij}$, $i > j$ are iid complex random variables with mean zero and unit variance, and the diagonal entries $\xi_{ii}$ are iid real variables, independent of the upper-triangular entries, with bounded mean and variance.
Unlike some definitions, Tao's definition makes no restriction on higher moments of $\xi_{ij}$ and in particular does not require the distribution of $\xi_{ij}$ to be symmetric. This suggests, to me, doing something like rescaling $Q$ and subtracting the mean to obtain $$ A^{n} = a M^{n} + q J^{n} $$ where $M$ is a Wigner matrix, $a$ and $q$ are appropriately chosen real constants and $J^{n}$ is a matrix of all ones $J_{ij} = 1$. If I have not made a mistake so far, then it seems I have stumbled into the realm of rank one deformations of Wigner matrices (since $J^{n}$ has rank one).
While I am not very familiar with the literature, it seems there is a lot known about deformations of Wigner matrices (but with varying definitions which are not always so general) by rank one matrices (but not necessarily of this form).
So my question is, what can be said about the distribution of the largest eigenvalue for this class of matrices? Numerically, I find behavior that appears to be very similar to results that have been proven for other closely related ensembles but I am not clear if they have been proven for this class. e.g. As I effectively increase $q$ there is a sharp transition in behavior where the largest eigenvalue $\lambda_{1}$ breaks away from the semicircle law and the distribution of the largest eigenvalue transitions from appearing to be Tracy-Widom like to something rather Gaussian. Please see the figures below. The first one shows histograms of the eigenvalues of ensembles of $100 \times 100$ matrices where $Q_{ij}\sim Tri(-1/2, m, 1/2)$ (triangular distribution with mode $m$ and support between -1/2 and 1/2) for different values of $m$. The second figure shows the first four moments of the distribution of the largest eigenvalue for this ensemble as a function of $m$.
