Supercritical multitype branching processes: the ancestral types of typical individuals

For supercritical multitype branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on non-extinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a `conceptual proof' (in the sense of Kurtz, Lyons, Pemantle, Peres 1997) of the continuous-time version of the Kesten-Stigum theorem.Comment: 23 pages, 1 figure; minor additions, added reference

Mean-Field Theory of the Potts Gas

International audienceWe consider a gas of classical particles in $\mathbb{R}^d$ having $q$ distinct colours, interacting via a mean-field Potts potential, and subject to an external field; a colour-independent molecular interaction of mean-field type is also admitted. In contrast to the usual lattice Potts model, the Potts gas exhibits the specific volume $v$ as a parameter, in addition to colour-ordering. The $v$-dependence of the Potts gas is studied in detail, and the complete phase diagram is derived. It turns out that, for $q\ge3$, the transition to colour-ordering implies a jump of density. For $q=2$, this transition is continuous but may become discontinuous under the influence of a suitable molecular interaction

Percolation and number of phases in the 2D Ising model

We reconsider the percolation approach of Russo, Aizenman and Higuchi for showing that there exist only two phases in the Ising model on the square lattice. We give a fairly short alternative proof which is only based on FKG monotonicity and avoids the use of GKS-type inequalities originally needed for some background results. Our proof extends to the Ising model on other planar lattices such as the triangular and honeycomb lattice. We can also treat the Ising antiferromagnet in an external field and the hard-core lattice gas model on $Z^2$.Comment: 22 pages. Further details on extensions. To appear in J.Math.Phys., special issue on `Probabilistic Methods in Statistical Physics', March 200