Consider a fixed real value $\sigma>0$. Let $A,Z$ be two independent $n\times n$ GOE matrices, and define $B=A+\sigma Z$. I am interested in finding a bound (possibly dependent on $n$) for the probability $$ \mathbb{P}(\exists i,j,\text{ s.t} \ \lambda_i(A)=\lambda_j(B)),$$ $\lambda_i(\cdot)$ denotes the $i$-th largest eigenvalue of a matrix. Essentially, I want to bound the probability that the spectra of $A$ and $B$ have a non-empty intersection. I suspect that this event has a low, or even zero, probability, but I am unsure how to tackle such a question. Any insights or suggestions on how to approach this problem would be greatly appreciated.
