Let $T$ be a uniformly random Standard Young Tableau (SYT) of shape $\lambda=(\lambda_1,\cdots,\lambda_k)$ with $|\lambda|=n$. Let $T_{ij}$ denote the value in box $(i,j)$. I'm interested in what can be said about correlations like $P(T_{ij}=x,T_{kl}=y)$. Very roughly speaking, if $|\lambda|=n$ is quite large and the $\lambda_i$'s are also large, then I would expect that $P(T_{ij}=x,T_{kl}=y)\approx P(T_{ij}=x)P(T_{kl}=y)$ for $(i,j)$ far apart from $(k,l)$ and for $x,y$ "not unreasonable" (see next part).
As a working example lets just take a square partition $\lambda=(n,n,\cdots,n)$ with $|\lambda|=n^2$. We know that such a partition has a well defined limit shape, so that if $x\approx an^2,y\approx bn^2$, then we expect $x,y$ to fall on limit curves, via what's more-or-less a semicircle distribution, i.e. being more likely to to fall near the middle of the limit curve than at the ends. My question is, on what scales do we know the error term $\delta$ of:
$$P(T_{ij}=x,T_{kl}=y)=P(T_{ij}=x)P(T_{kl}=y)+\delta_{ij,kl}(x,y)?$$
In other words, what does $\delta$ look like for $x,y=\Omega(n^2)$ and in particular when $|x-y|=o(n^2)$ or $|i-k|^2+|j-l|^2=o(n)$? Specifically, what is the order of $\delta$ in terms of $n$ and $(i,j),(k,l)$?. I'm purposely leaving a lot of room for interpretation because I'm guessing overall these are very difficult questions to answer for all possible cases of $x,y$.
