Let $n$ be a positive integer and consider the probability density $f_n$ on $\mathbb R_+$ given by $f_n(z):=\int J_n(u,u+z)du$, where $J$ is a probability density on $\mathbb R^2$ given by $J_n(u,v):=\dfrac{n(n-2)}{2\pi}e^{-(u^2+v^2)/2}\Phi(u)^{n/2}1_{u \le v}$, and $\Phi$ is the standard normal CDF. Let $Z_n$ be the random variable on $\mathbb R_+$ with probability density $f_n$. Thanks to this post, we know that $Z_n$ is distributionally equal to the difference between the largest and the second largest value in an iid sample of size $n$ from the standard normal distribution $N(0,1)$.
Question. What are good non-asymptotic lower- and upper-bounds for $\mathbb P(Z_n \ge z)$ in terms of $n$ and $z$ alone ?
