Let $(X_1,X_2,\ldots)$ be a sequence of i.i.d. random variables. It is known that if these random variables are distributed uniformly on the unit interval, then applying the RSK algorithm to this sequence (and looking at the recording tableau) gives the (infinite) Plancheral measure on Young Tableaux. Restricting the above sequence to length $n$ to give the Plancheral measure on partitions of $n$. The proof is straightforward as the sequence of $X_i$ induce uniformly random permutations and RSK follows through.
Here's my question: what is known in the case of other distributions for the $X_i$? I suppose one can equivalently say what is known for non uniformly random permutations but I'd like to stick to the former.
In particular, is anything known about the resulting limit shape of the tableau like the result of Logan-Shepp-Vershik-Kerov? I did some simulations with other distributions and it seems the limit shape is the same! Here's a picture of the usual Plancheral limit shape: