# Eigenvalue difference of $2 \times 2$ real symmetric random matrix

Let $H:= \begin{pmatrix} h_{11} & h_{12} \\ h_{12} & h_{22} \end{pmatrix}$ be a $2 \times 2$ real symmteric random matrix with continuously distributed independent entries and real eigenvalues $\lambda_1$ and $\lambda_2$. I want to show that $P(|\lambda_1-\lambda_2| \le \epsilon) \sim \epsilon^2$, in the sense that the probability scales like $\epsilon^2$ for small $\epsilon$. My idea was first to compute the eigenvalue difference in terms of the matrix elements, which results, if I'm not mistaken, in $|\lambda_1-\lambda_2|=\sqrt{(h_{11}-h_{22})^2+4h_{12}^2}$, so one should be able to argue that this difference is $0$ iff $h_{11}-h_{22}$ and $h_{12}$ both are $0$. Does this observation help to obtain the desired scaling property? If yes, how can I proceed from here?

We start by using your formula for the difference $\left|\lambda_1-\lambda_2\right| = \sqrt{\mathrm{Tr}^2H - 4\det H} = \sqrt{\left(h_{11}-h_{22}\right)^2+4h_{12}^2}$. One can now use independence of rv's to argue $$P\left(\left|\lambda_1-\lambda_2\right|\leq \varepsilon\right) \leq P\left(\left|h_{11}-h_{22}\right|\leq \varepsilon\right)P\left(\left|h_{12}\right|\leq\varepsilon/2\right).$$
Remark: You can also show that $P\left(\left|\lambda_1-\lambda_2\right|\leq \varepsilon\right) \geq P\left(\left|h_{11}-h_{22}\right|\leq \varepsilon/2\right)P\left(\left|h_{12}\right|\leq\varepsilon/4\right).$
Now let $\rho_{11},\rho_{22},\rho_{12}$ be corresponding probability density functions, then $$p_1:=P\left(\left|h_{12}\right|\leq\delta\right) = \int\limits_{-\delta}^{\delta} \rho_{12}\left(x\right) \,dx,$$ $$p_2:=P\left(\left|h_{11}-h_{22}\right|\leq \delta\right) = \int\limits_{-\infty}^{\infty}\rho_{11}\left(x\right)\int\limits_{x-\delta}^{x+\delta} \rho_{22}\left(y\right)\,dy\,dx.$$ And now one can see that the final estimate heavily relies on properties of distributions. If say, $\rho_{22},\rho_{12}$ are uniformly bounded from above and below by positive constants, i.e. $0<c\leq \rho_{22}(x),\rho_{12}(x)\leq C <\infty$, then $2c\delta\leq p_1,p_2\sim 2C\delta$. However if densities are bounded from above only, you can get only estimates of form $p_1,p_2\leq C\delta$. Finally, if you have no prior information on the distributions of matrix elements, you can get an asymptotic result different from the one you expect.