Universal behaviour of 3D loop soup models
Abstract
These notes describe several loop soup models and their {\it universal behaviour} in dimensions greater or equal to 3. These loop models represent certain classical or quantum statistical mechanical systems. These systems undergo phase transitions that are characterised by changes in the structures of the loops. Namely, longrange order is equivalent to the occurrence of macroscopic loops. There are many such loops, and the joint distribution of their lengths is always given by a {\it PoissonDirichlet distribution}. This distribution concerns random partitions and it is not widely known in statistical physics. We introduce it explicitly, and we explain that it is the invariant measure of a meanfield splitmerge process. It is relevant to spatial models because the macroscopic loops are so intertwined that they behave effectively in meanfield fashion. This heuristics can be made exact and it allows to calculate the parameter of the PoissonDirichlet distribution. We discuss consequences about symmetry breaking in certain quantum spin systems.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 arXiv:
 arXiv:1703.09503
 Bibcode:
 2017arXiv170309503U
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 31 pages, 11 figures. Notes prepared for the 6th Warsaw School of Statistical Physics, held from 25 June to 2 July 2016 in Sandomierz, Poland