Normalizing Entries In Defining Random Matrices (Wigner Matrix)

Question

In the definition of Wigner Matrix (a certain type of random Matrices) we take to independent family of i.i.d zero mean distributions $\{Z_{i,j}\}_{1<i<j}$ and $\{Y_{i}\}_{1\leq i}$ and then the entries of Wigner Matrix $X$ define as below:

$$\forall i,j; 1\leq i<j\leq N : X_N(i,j) = Z_{i,j}/\sqrt{n}$$ $$\forall i,j; 1\leq j<i\leq N : X_N(i,j) = Z_{j,i}/\sqrt{n}$$ $$\forall i; 1\leq i \leq N : : X_N(i,i) = Y_{i}/\sqrt{n}$$

After this task we will have a self-adjoint matrix that its eigenvalues are all real and we have Wigner's Semicircle Law about it.

My question is this: Is there any intuition or short proof that we should normalize the entries with $\frac{1}{\sqrt{N}}$ for controlling eigenvalues from being large.

Answer 1

On the one hand, you can compute the average of the trace of $X^2$ by using the eigenvalues, $\langle {\rm Tr}(X^2)\rangle=\sum_{i}\langle \lambda_{i}\rangle$. If the eigenvalues must be $O(1)$, this must be $O(N)$.

On the other hand, you may compute this average using matrix elements, $\langle {\rm Tr}(X^2)\rangle=\sum_{i,j}\langle X_{i,j}X_{j,i}\rangle$. If you don't normalize the entries with $1/\sqrt{N}$, this won't be $O(N)$ as required.