Asymtotics of representations of classical Lie groups and probabilistic problems

Asymptotic representation theory studies representations of groups of growing size (while in the classical theory a group is usually fixed). It turns out that the arising in this context problems are close to the number of combinatorial and probabilistic subjects such as random matrices, free probability, random increasing subsequences, random plane partitions. I will give a short review of these areas with a special emphasis on connections between them.