What's Happening in Fine Hall

The interchange process \sigma_T is a random permutation valued process on a graph evolving in time by transpositions on its edges at rate 1.  On Z^d, when T is small all the cycles of the permutation \sigma_T are finite almost surely.  In dimension d \geq 3 infinite cycles are expected when T is large.  The cycles can be interpreted as a random walk which interacts with its past and we give a multi-scale proof establishing transience of the walk (and hence infinite cycles) when d\geq 5.