For a Boolean function $f\colon\{-1,1\}^n \to \{-1,1\}$, the influence of the $i$th variable is defined as $$ \operatorname{Inf}_i[f] \stackrel{\rm def}{=} \Pr_{x\sim\{-1,1\}^n}[ f(x) \neq f(x^{\oplus i})] $$ where $x^{\oplus i}$ is the string obtained by flipping the $i$th bit of $x$. The minimum influence of $f$ is then $$\operatorname{MinInf}[f] \stackrel{\rm def}{=} \min_{i\in[n]}\operatorname{Inf}_i[f].$$
Given a parameter $p\in[0,1]$, we choose a $p$-random function $f$ by choosing its value on each of the $2^n$ inputs independently at random to be $1$ with probability $p$, and $-1$ with probability $1-p$. Then, it is easy to see that, for every $i\in[n]$ $$ \mathbb{E}_{f}[\operatorname{Inf}_i[f]] = 2p(1-p) $$ and a fortiori $$ I_n(p) \stackrel{\rm def}{=}\mathbb{E}_{f}[\operatorname{MinInf}[f]] \leq 2p(1-p). $$
My question is:
Is there an asymptotically (with regard to $n$) tight expression for $I_n(p)$? Even for $p=\frac{1}{2}$, can we get such an expression?
Specifically, I do care about the low order terms, i.e. I'd be interested in an asymptotic equivalent for the quantity $2p(1-p)-I_n(p)$.
(The next question, but which is subordinate to the first, is whether one can also get good concentration bounds around this expectation.)
By Chernoff bounds one can also show that each $\operatorname{Inf}_i[f]$ has good concentration, so that by a union bound we get (if I did not mess up too badly) $$ \frac{1}{2} - O\left(\sqrt{\frac{n}{2^n}}\right) \leq I_n\left(\frac{1}{2}\right) \leq \frac{1}{2} $$ but this is most likely loose on the lower bound (due to the union bound) and definitely on the upper bound. (I am in particular looking for an upper bound strictly less than the trivial $\frac{1}{2}$).
Note that one of the issues in doing that, besides taking the minimum of $n$ identically distributed random variables (the influences), is that these random variables are not independent... although I do expect their correlation to decay "pretty fast" with $n$.
(For what it's worth, I have computed explicitly the first few $I_n(1/2)$'s up to $n=4$, and have run simulations to estimate the following ones, up to $n=20$ or so. Not sure how helpful this could be, but I can include that once I am back to my office.)