Given a $n\times n$ symmetric random matrix whose diagonal elements are all fixed as $1$. In addition, there are $k$ $1$s will be randomly scattered in upper triangular (of course, the corresponding places in the lower-triangle will be filled with $1$, and $2k < n^2-n$). All other elements are independent uniform random variables over $[0,1]$.
Is there known bound (lower and upper) for the largest eigenvalue of such random matrices?
If there is not, any suggestion of possible method (I can think of using Gershgorin circle) or reference to related materials is very much appreciated.
Gershgorin circle could help with the upper bound. For example, if we assume all those $1$s are in the same row, then we should be able to find the probabilistic bound for this case with Irwin–Hall distribution; but I currently have trouble dealing with the "randomly scattered" $1$s.
I am not familiar with the random matrix theory. I am not sure if there is anything from it can help this.