TITLE: Random RSK and Plancherel Measure
QUESTION [9 upvotes]: 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:
(source)

REPLY [7 votes]: Another large family of distributions that contains both $U[0,1]$ inputs (leading to the usual Plancherel measure) and $U\{1,2,\ldots,d\}$ (as in Ryan O'Donnell's answer) as special cases is the following. Let $(\alpha, \beta, \gamma)$ be an element of the Thoma simplex, i.e., a triple where
$\gamma\in [0,1]$ and $\alpha$ and $\beta$ are vectors
$$ \alpha = (\alpha_1, \alpha_2, \ldots) \ \ \textrm{ with }\alpha_1\ge\alpha_2\ge \ldots \ge 0,
$$
$$ \beta = (\beta_1, \beta_2, \ldots) \ \ \textrm{ with }\beta_1\ge\beta_2\ge \ldots \ge 0,
$$
and such that 
$$
\sum_n \alpha_n + \sum_n \beta_n + \gamma = 1.
$$
We can associate with $(\alpha, \beta, \gamma)$ a probability distribution of a random variable $X$ that is supported on $(0,1) \cup \{1,2,3,\ldots\} \cup \{-1,-2,-3,\ldots\}$, defined by
$$ \mathbb{P}(X \in (a,b)) = \gamma (b-a) \qquad (0<a<b<1), 
$$
$$ \mathbb{P}(X = n) = \alpha_n \qquad (n=1,2,\ldots), 
$$
$$ \mathbb{P}(X = -n) = \beta_n \qquad (n=1,2,\ldots). 
$$
If we now take a sequence of i.i.d. random variables $X_1,X_2,\ldots$ with the same distribution as $X$ and apply the RSK algorithm to them, the resulting random infinite Young tableau has a very natural probability distribution studied by Vershik and Kerov, and supposedly related to the representation theory of the infinite symmetric group (a subject I know nothing about). 
Note, however, that requires a modified version of RSK in which the rules for the insertion tableau are different: the rows and the columns of the insertion tableau are weakly increasing, but it is not allowed to have the same negative integer twice in the same row and it is not allowed to have the same positive integer twice in the same column.
This result was proved by Sergei V. Kerov and Anatol M. Vershik (SIAM J. Algebraic Discrete Methods, 7(1):116–124, 1986; available here)
Some additional consequences are discussed in the paper "Robinson-Schensted-Knuth algorithm, jeu de taquin, and Kerov-Vershik measures on infinite tableaux" by Piotr Śniady (SIAM J. Discrete Math. 28 (2014), 598–630; available at these links: journal version, arXiv version.)