Let us consider a deck of n cards. Given a graph on n vertices, we shuffle our cards by each time choosing a random edge (i,j) of the graph and interchanging the cards at locations i and j. This process and some immediate generalizations thereof are called the interchange process. Note that at each stage we get a permutation of the cards, and therefore our process is a random walk on the symmetric group Sn.
The interchange process has a deep connection to the representation theory of the symmetric group, making it a fascinating meeting point of probability and algebra.
In this talk we will explain this connection and survey some (relatively) old and new results: The proof of the Aldous conjecture by P. Caputo, T. Liggett and T. Richthammer; Some joint works of the speaker with G. Kozma and D. Puder, and some generalizations by A. Bristiel, P.Caputo, F. Cesi and S.Ghosh.
