Random forests and fermionic field theories

mercredi 29 oct. 2025, 13:30 → 17:30 Europe/Paris

Amphithéâtre Gaston Darboux (IHP - Bâtiment Perrin)

The Seed seminar of mathematics and physics is a seminar series that aims to foster interactions between mathematicians and theoretical physicists, especially among young researchers. It is structured into three-month thematic periods, the fall 2025 one being on Random forests and fermionic field theories.

We open this thematic trimester with an in-person kick-off event at the Institut Henri Poincaré with contributions from Andrea Sportiello, Sonia Boulal and Julien Weibel.

Registration for attending the event in person is free but mandatory, see Registration in the indico menu.

If you cannot attend the event in person but are interested in following the talks online, please subscribe here to the Seed seminar mailing list, on which Zoom links will be shared for this event and future ones.

1 Grassmann Calculus for the combinatorics of Spanning Trees and Forests

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.

Orateur: Andrea SPORTIELLO (CNRS & LIPN, Université Paris 13)

14:30 Pause café

2 Conditioned Marked Galton-Watson trees

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.

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

15:50 Pause café

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

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.

Orateur: Julien WEIBEL (Inria Paris)