Consider a random mapping $f:\{0,1\}^n \to \{0,1\}^n$, .i.e, a function such that for each $x \in \{0,1\}^n$, $f(x) \in \{0,1\}^n$ is chosen uniformly at random.
My question is what would the fourier expansion of $f$ look like? Could something be said of the concentration of the fourier coefficients at certain weights?
Additionally, if we create a "spike" in the function by choosing some $y_0 \in \{0,1\}^n$ and a subset $S \subset \{0,1\}^n$ and then set $f(x) = y_0$ for all $x \in S$ (when $S$ is relatively "large") then does this translate to any interesting behavior in the fourier expansion?
(Please note that this question is based on a question in math stackexchange https://math.stackexchange.com/questions/4719971/fourier-transform-of-random-mapping)
