Correlation of Max and Min

The Problem

Let $X_1,X_2,\dots ,X_n$ be i.i.d. $\text{Uniform}(0,1)$ random variables. Define $M_n=\max (X_1,\dots ,X_n)$ and $m_n=\min (X_1,\dots ,X_n)$.

What is the correlation $\rho (M_n,m_n)$ between the maximum and the minimum? How does it behave as $n\to \infty$?