Grassmann Calculus for the combinatorics of Spanning Trees and Forests

• Andrea SPORTIELLO (CNRS & LIPN, Université Paris 13)

Room: Amphithéâtre Gaston Darboux In this talk we will make a survey of how techniques of ``Grassmann Calculus'', that is, integration of expressions involving anticommuting variables, provide fermionic analogues of Gaussian integration, Wick's Theorem and perturbative field theory. These techniques are specially fruitful for describing certain combinatorial models in Statistical Mechanics, namely $n=2$ Loop Models, Spanning Trees, and Spanning Forests. If the time permits, we will also show how the model of Spanning Forests, in its Grassmann-variable formulation, has a hidden $\mathrm{OSp}(1|2)$ supersymmetry, that, by the Parisi--Sourlas mechanism, implies that it must be in the same universality class of the $O(n)$ loop model in the analytic continuation $n\to -1$. Mostly based on (old) works in collaboration with S. Caracciolo and A.D. Sokal.

Conditioned Marked Galton-Watson trees

• Sonia BOULAL (Institut Denis Poisson, Université d’Orléans)

Room: Amphithéâtre Gaston Darboux We consider a Galton–Watson tree in which each node is independently marked, with a probability that depends on its number of offspring. We give a complete picture of the local convergence of critical or subcritical marked Galton–Watson trees, conditioned on having a large number of marks. In certain cases, the limit is a randomly marked tree with an infinite spine, known as the marked Kesten tree. In other cases, the local limit is a randomly marked tree with a node having infinitely many children. This corresponds to the so-called marked condensation phenomenon. Joint work with Romain Abraham and Pierre Debs.

Probability-graphons: limits for sequences of large dense weighted graphs

• Julien WEIBEL (Inria Paris)

Room: Amphithéâtre Gaston Darboux Networks appear naturally in a wide variety of context, including for example: biological networks , epidemics processes, electrical power grids and social networks. Most of those problems involve large dense graphs, that is graphs that have a large number of vertices and a number of edges that scales as the square of the number of vertices. Those graphs are too large to be represented entirely in the targeted applications. The idea is then to go from a combinatorial representation given by the graph to an infinite continuum representation: the graphons studied by Lovász and his co-authors. In this talk, I will present a joint work with Romain Abraham and Jean-François Delmas on probability-graphons, which are measure-valued graphons that generalises graphons to the case of weighted graphs and decorated graphs. I will explain how probability-graphons are used to define random weighted graph models, define a distance that control convergence for those sampled graphs, and give some topological results on this distance.