The probability distribution of the spacing $\delta_N=\lambda_N-\lambda_{N-1}$ of the eigenvalues $\lambda_N$ and $\lambda_{N-1}$ at the edge of the spectrum decays exponentially for $\delta_N\gg N^{-2/3}$, with a decay rate that is independent of $N$. So no matter how small $\epsilon$, you can always find a $c$ such that $P(N^{2/3}\delta_N\le c)\ge 1-\epsilon$.

I agree with the "robot" ChatGPT in the deleted answer that $c$ will depend on $\epsilon$, but I do not agree that $c$ will depend on $N$. The scaling with $N$ is fully contained in the mean level spacing $N^{-2/3}$ for large $N$ (Tracy-Widom law).

The probability distribution of the spacing $\delta_N=\lambda_N-\lambda_{N-1}$ of the eigenvalues $\lambda_N$ and $\lambda_{N-1}$ at the edge of the spectrum decays exponentially for $\delta_N\gg N^{-2/3}$, with a decay rate that is independent of $N$. So no matter how small $\epsilon$, you can always find a $c$ such that $P(N^{2/3}\delta_N\le c)\ge 1-\epsilon$.

I agree with the ChatGPT bot in the deleted answer that $c$ will depend on $\epsilon$, but I do not agree that $c$ will depend on $N$. The scaling with $N$ is fully contained in the mean level spacing $N^{-2/3}$ for large $N$ (Tracy-Widom law).