Gibbs fragmentation trees

We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous' beta-splitting model, which has an extended parameter range $\beta>-2$ with respect to the ${\rm beta}(\beta+1,\beta+1)$ probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter Poisson--Dirichlet models for exchangeable random partitions of $\mathbb {N}$, with an extended parameter range $0\le\alpha\le1$, $\theta\ge-2\alpha$ and $\alpha<0$, $\theta =-m\alpha$, $m\in \mathbb {N}$.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ134 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Universal behaviour of 3D loop soup models

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, long-range 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 Poisson-Dirichlet 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 mean-field split-merge process. It is relevant to spatial models because the macroscopic loops are so intertwined that they behave effectively in mean-field fashion. This heuristics can be made exact and it allows to calculate the parameter of the Poisson-Dirichlet distribution. We discuss consequences about symmetry breaking in certain quantum spin systems.Comment: 31 pages, 11 figures. Notes prepared for the 6th Warsaw School of Statistical Physics, held from 25 June to 2 July 2016 in Sandomierz, Polan