Let $(X\_1, X\_2, \dots, X\_n)$ be i.i.d. ${\cal N}(0,1)$. We construct a random circulant matrix $M$:


$$M = \frac{1}{\sqrt n}\begin{pmatrix}X\_1 &X\_2 &X\_3 \dots &X\_n\\ X\_n &X\_1 & X\_2 \dots &X\_{n-1}\\ \vdots &\vdots &\vdots &\vdots\\X\_2 &X\_3 &X\_4 \dots &X\_1\end{pmatrix}.$$


My questions are the following:


1. Is $M$ "almost orthonormal" in a precise probabilistic sense?
2. Related to above, is it possible to upper-bound the largest possible dot product between any two rows of $M$ by a suitably small $\epsilon\_n$?


Note that this says that $MM^T$ is close to the identity matrix $I\_{n \times n}$, as we are bounding the off-diagonal entries of $MM^T$ by $\epsilon\_n$, while the diagonal entries are close to $1$. 



Are random circulant matrices almost orthonormal?
The diagonal elements of $P=\frac{1}{N}MM^T$, like
$$P\_{11}=\frac{1}{N}\sum\_{i=1}^NX\_i^2,$$
satisfy $ \langle P\_{11}\rangle=1$ and $ \langle P\_{11}^2\rangle=1+2/N$ (variance decreases like $N^{-1}$).


On the other hand, off-diagonal elements like
$$P\_{12}=\frac{1}{N}\sum\_{i=1}^{N}X\_iX\_{i+1} $$
satisfy $ \langle P\_{12}\rangle=0$ and $ \langle P\_{12}^2\rangle=1/N$ (variance also decreases like $N^{-1}$).


In this sense I would say $P$ is close to the identity.


