Let $\mathbf{u} =\begin{bmatrix}u_1 & u_2 & \dots & u_N \end{bmatrix}^T$ and $\mathbf{v} = \begin{bmatrix} v_1 & v_2 & \dots & v_N\end{bmatrix}^T$. All the elements of $\mathbf{u}$ and $\mathbf{v}$ are complex Gaussian random variables with zero mean and variance $\frac{1}{N}$ and $N$ is large. Also let $x_1, x_2 \in \{-1,1\}$ where $x_1$ a can be $-1$ or $1$ with equal probability $(p=0.5)$ and similaraly $x_2$.
Define $d^2$ as the square of the euclidean distance between $\mathbf{u}x_1 \text{and } \mathbf{v}x_2$: $$d^2=\mathbf{|u}x_1-\mathbf{v}x_2|^2 = |u_1x_1-v_1x_2|^2 + \dots +|u_Nx_1-v_Nx_2|^2 \\ \text{let} ~~d_\min = \min(d^2)~~ \text{what is} ~~ E(d_\min) \text{?}$$.
There are $4$ combinations for $(x_1, x_2)$: $(1,1),(1,-1),(-1,1),(-1,-1)$ and therefore $4$ combinations for $d^2$ $$d_1^2=|u_1-v_1|^2 + \dots +|u_N-v_N|^2 \\ d_2^2=|u_1+v_1|^2 + \dots +|u_N+v_N|^2 \\ d_3^2 = |-u_1-v_1|^2 + \dots + |-u_N-v_N|^2 \\ d_4^2 = |-u_1+v_1|^2 + \dots +|-u_N+v_N|^2$$ Since $u_i$ and $v_i$ are complex Gaussian, $|u_i-v_i|$ is Rayleigh and $|u_i-v_i|^2$ is going to be exponential. $N$ is large here and since all terms $|u_i-v_i|^2$ are independent I can use the central limit theorem and say that $d_1^2,d_2^2,d_3^2,d_4^2$ are all Gaussian. I can use the CDF method to calculate the distribution of $d_\min$ and then calculate the average minimum distance but $d_1^2,d_2^2,d_3^2,d_4^2$ are not independent and I don't know how to proceed. Any help/guidance is greatly appreciated.