  These are notes for a mini-course of 3 lectures given at the St. Petersburg
School in Probability and Statistical Physics (June 2012). My aim was to
explain, on the example of a particular model, how ideas from the
representation theory of big groups can be applied in probabilistic problems.
  The material is based on the joint paper arXiv:1009.2029 by Alexei Borodin
and myself; a broader range of topics is surveyed in the lecture notes by
Alexei Borodin and Vadim Gorin arXiv:1212.3351.
