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Given a $n\times n$ symmetric random matrix whose diagonals 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 $[0,1]$.

Help with the bound (lower and upper) for the largest eigenvalue of such random matrices.

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.

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Your question is poorly written; consider, for example, the largest eigenvalue denoted by $\lambda_{n,k}$; what do you want to know ? $E(\lambda_{n,k})$ ? $V(\lambda_{n,k})$ ? $\sup(\lambda_{n,k})$ ?

We consider the last question; note that it is not a random problem. Clearly, $\lambda_{n,k}\leq n$ and $\lambda_{n,n(n-1)/2}=n$.

In particular, if the uniform random variables over $[0,1]$ are all $1$, then $\lambda_{n,k}=n$.

On the other hand, if the uniform random variables over $[0,1]$ are all $0$, then $\sup(\lambda_{n,k})$ is reached when the up and left square submatrix is full of $1$.

For example $\sup(\lambda_{10,10})=5$ and is reached (between others) by the matrix $diag(U,I_5)$ where $U\in M_5$ is the matrix of $1$.

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