Daniel Ueltschi

On spin systems, loop representations, and random partitions

Some classical and quantum spin systems (such as the Heisenberg models) are equivalent
to probabilistic models of random loops. Looking at the lengths of the loops, we get measures
on random partitions. It is conjectured that for quantum systems in dimensions 3 or greater,
the measure on random partitions is always a Poisson-Dirichlet distribution.

I will explain what is known and what is expected in this context. I will also address the
question of characterising random partition through “random colouring”. This is indirectly related
to symmetries of the spin systems.

Based on joint works with C. Benassi, J. Björnberg, C. Mailler, P. Mörters.