For some further results of this nature, see Exercise 5.64 of Enumerative Combinatorics, vol. 2. This exercise deals with the uniform distribution on (0,1)-matrices or $(-1,1)$-matrices, but the arguments can be carried over to other distributions where the matrix entries are i.i.d. The proofs are similar to the argument in David Speyer's comment.
5.64. a. [2+] Let $\mathcal D_n$ be the set of all $n\times n$ matrices of $+1$'s and $-1$'s. For $k\in\mathbb P$ let \begin{align*} f_k(n)&= 2^{-n^2} \sum_{M\in\mathcal D_n} (\det M)^k \\ g_k(n)&= 2^{-n^2} \sum_{M\in\mathcal D_n} (\operatorname{per} M)^k, \end{align*} where $\operatorname{per}$ denotes the permanent function defined by $$\operatorname{per}(m_{ij})= \sum_{n\in\mathfrak{S}_n} m_{1,\pi(1)} m_{2,\pi(2)} \dots m_{n,\pi(n)}.$$ Find $f_k(n)$ and $g_k(n)$ explicitly when $k$ is odd or $k=2$.
b. [3-] Show that $f_4(n)=g_4(n)$, and show that $$\sum_{n\ge 0} f_4(n) \frac{x^n}{n!} = (1-x)^{-3} e^{-2x}. \tag{5.120}$$ HINT. We have $$\sum_m (\det M)^4 = \sum_M \left(\sum_{\pi\in\mathfrak S_n} \pm m_{1,\pi(1)}\dots m_{n,\pi(n)}\right)^4.$$ Interchange the order of summation and use Exercise 5.63.
c. [2+] Show that $f_{2k}(n)<g_{2k}(n)$ if $k\ge 3$ and $n\ge 3$.
d. [3-] Let $\mathcal D'_n$ be the set of all $n\times n$ 0-1 matrices. Let $f'_k(n)$ and $g'_k(n)$ be defined analogously to $f_k(n)$ and $g_k(n)$. Show that $f'_k(n)=2^{-kn} f_k(n+1)$. Show also that \begin{align*} g'_1(n) &= 2^{-n} n!\\ g'_2(n) &= 4^n n!^2 \left(1+\frac1{1!}+\frac1{2!}+\dots+\frac1{n!}\right) \end{align*}
