Collecting stones in n buckets There are $n$ stones distributed in $n$ buckets (initially one stone in each bucket). At each step the content of each bucket is put in a random bucket, chosen independently out of a set of $n$ new empty buckets. Doing this the groups of stones in the differents buckets may merge together.
Now, after $i$ steps we consider the non-empty groups of stones, and I was wondering if there is any information about the (two dimensional) distribution (giving a probability of having at least $\geq b$ buckets with $\geq k$ stones, say), in particular if the problem was studied for big $n$ and big $i$ relative to $n$. Was this model studied before?
The problem arises from simulations of discretized chaotic dynamical systems, where this model (could) provide a comparison model, for long time behaviour and fine discretization.
 A: See A balls-and-colours problem and Another colored balls puzzle although those don't talk about the two-dimensional distribution. These suggest looking at the count of pairs of pebbles in different buckets.
For $a \ne b$ to be sent to different places after $i$ steps, it must be that on each step, their buckets are emptied into different buckets. The probability of that is $\left(\frac{n-1}{n}\right)^i$. 
Let the number of pairs of pebbles $a \ne b$ sent to different places after $i$ steps be $P_i$. $E[P_{cn}] ={n \choose 2} \left(\frac{n-1}{n}\right)^{cn} \sim {n \choose 2}e^{-c}$. 
If two pebbles are not in the same bucket, then there are at least n-1 pairs of pebbles in different buckets. That means the probability that there are pebbles in different buckets after $cn$ steps is at most $E[P_{cn}]/(n-1) = \frac{n}{2} e^{-c}$. So, by $(1+o(1))n \log n$ steps, the probability that all pebbles are in the same bucket approaches $1$. (The data collected by Aaron Meyerowitz suggests that this might not be sharp.)
