Given two random sets $A,B$ in a finite field (say $x\in A$ independently and with probability $1/2$), what is known about the additive energy $E(A,B)=|\{(a,a',b,b')\in A\times A\times B\times B: a+b=a'+b'\}|$?


Equivalently, what is the distribution of the random variable 
$\|1\_A\*1\_B\|\_2$? I'm mostly interested in asap (as sharp as possible) upper bounds for the probability of it being large.



Additive energy of random sets
Expanding on my earlier comment, the concentration has to be really quite good: at least exponential in $N$, the order of the additive group. (Edit: and you can't get better concentration because the state space is only exponentially large.)


Recall the following variant of the Hoeffding-Azuma martingale concentration inequality. Let $\Omega = \Omega\_1 \times \cdots \times \Omega\_n$ be a product space, and let $X$ be a random variable on $\Omega$ such that $X(\omega)$ changes by at most $k$ if we change a single coordinate of $\omega$. Then $$\mathbb{P}(X \geq (1+t)\mu) \leq \exp (-2t^2\mu^2/nk^2),$$ where $\mu=\mathbb{E}(X)$. 


Apply this with $n=2N$, $\Omega$ the space of coin flips determining membership of each element of the group in $A$ and $B$, and $X=E(A,B)$. We have $\mu = N^3/16$ and $k = O(N^2)$ (as each element of the group is in $O(N^2)$ additive quadruples), which gives $$\mathbb{P}(X \geq (1+t)\mu) \leq \exp(-ct^2N),$$ for some absolute constant $c$.


