Poisson theorems for Robinson-Schensted-Knuth algorithm

We investigate a number of asymptotic questions related to Robinson-Schensted-Knuth algorithm applied to a random input and show that the answer for each of them is given by the Poisson process. The first problem concerns the growth of the bottom rows of the Young diagram which is subject to Plancherel growth process; we extend the result of Aldous and Diaconis to more than just one row. The second problem concerns the shape of the bumping route (in the vicinity of the $y$-axis) when a specified number is inserted into a large Plancherel-distributed tableau. (This is a joint work with Łukasz Maślanka and Mikołaj Marciniak)
Additional material to the lecture will be available (one week before the talk) at http://psniady.impan.pl/Poisson