Poisson limit theorems for the Robinson–Schensted correspondence and for the multi-line Hammersley process

We consider the Robinson–Schensted–Knuth algorithm applied to a random input and study growth of bottom rows corresponding Young diagrams. prove multidimensional Poisson limit theorem for resulting Plancherel process. In this way we extend result Aldous Diaconis more than just one row. This can be interpreted as convergence multi-line Hammersley process its stationary distribution which is given by collection independent point processes.