Mathematics Institute, University of Warwick

Phase transitions for quantum spin lattice models can be shown to be
linked with the
existence of large cycles for random interchanges of sites of the
corresponding lattice.
While for complete graphs the random interchanges are well understood
(cards shuffling), the emergence of large cycles for lattices is an
entirely open problem.
The case of hypercube is a first step in this direction.
Based on a joint paper with P. Milos and D. Ueltschi.