For an application quite alien to probability theory, I'd like to have a kind of measure concentration estimate, in the following spirit. Suppose that to every $1\le i,j\le n$ there corresponds a zero-mean random variable $X_{ij}$, all of them being identically distributed and taking values in $[-1,1]$. Let $S$ denote the normalized sum of all these $n^2$ random variables: $S:=(X_{11}+\dotsb+X_{nn})/n$. I want to conclude that under some week dependence assumption (to be stated immediately), one has $$ {\mathsf P}(|S|>\tau) <C e^{-\gamma\tau},\quad \tau\ge 1, $$ with some absolute constants $C,\gamma>0$. (I originally hoped to prove that this probability is $O(e^{-\gamma\tau^2})$, but Carl noticed that this is too optimistic.) Indeed, I may be happy with some slightly weaker estimate, or even with an estimate like ${\mathsf E} |S|^p\le(Mp)^p$ with an absolute constant $M$.)

Now, the weak dependence assumption just mentioned is that, viewing the $X_{ij}$ as the entries of a matrix, if several of them are dependent, then one can create a closed aligned loop in the matrix with the corresponding entries being the vertices of the loop. In particular,

- the variables are independent pairwise and in triples;
- any four of them are independent, unless their indices form a rectangle in the matrix;
- any system of the variables no two of which are in the same column or in the same row is independent.

In general, I wonder whether this weak dependence assumption has ever been studied, and what conclusions can be drawn from it. Thanks in advance for any suggestions or pointers!