I'm trying to find theorems regarding random variables derived from sampling permutations, specifically concentration bounds.

As an example, let $X_i$ be the $\{0,1\}$-random variable that represents whether a uniformly random permutation $\sigma \in S_m$, has that $\sigma(i)$ is even (assuming permutations in $S_m$ are taken as bijections from $\{1, \ldots, m\}$ to itself)

Note that then the $X_i$ are identically distributed, but not independent, because each $X_i$ essentially 'samples without replacement' from the set $\{1,\ldots,m\}$

Then it's easy to see that (say) for $n = \frac{m}{2}$, $Pr[X_1 + X_2 + \ldots + X_n = n] = \frac{n!}{m!}$ which is asymptotically small.

More generally, given some set of functions $f_i: S_m \rightarrow \{0,1\}$, let $Y_i$ be a $\{0,1\}$-random variable that represents whether a uniformly random permutation $\sigma \in S_m$ has $f_i(\sigma) = 1$.

Then it seems plausible, given some reasonable conditions on the $f_i$ (for example, maybe $Pr[Y_i = 1] = 1/c$ for some constant $c$, or $Pr[Y_i + Y_j = 2] < 1/c$) that we can get some kinds of concentration bound for $Y_1 + \ldots + Y_n$

The best I can think of is to use the above two conditions on the $f_i$'s to calculate the expected value and variance of $Y_1 + \ldots + Y_n$ then invoke Chebyshev's inequality.

However, I can't believe that there's not some additional conditions that would lead to a better bound. Considering the "$\sigma(i)$ is even" example above, even though the variables are not mutually independent (or even pairwise independent) we still have that $X_1 + \ldots X_n$ is strongly concentrated around its expectation (in some vague sense).

Is there some way to interpret random variables like $X_1, \ldots, X_n, Y_1, \ldots, Y_n$, as 'nice' samples of a permutation distribution, as retaining some of the properties of independence that might lead to concentration?

This isn't my area so I might not be asking in the best way, so part of my question is also "what's the right way to ask this question?" :) Thanks a lot.