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Quatrième rencontre de probabilités intégrables
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Europe/Paris
Salle Pierre Grisvard (IHP - Bâtiment Borel)
Salle Pierre Grisvard
IHP - Bâtiment Borel
Description
La quatrième rencontre de probabilités intégrables aura lieu le vendredi 3 avril 2026 à l'Institut Henri Poincaré.
Titres et résumés
Mini-cours (2x1h) par Ariane Carrance et Baptiste Louf :The Ising model on combinatorial maps, in every genus
Résumé:
In this mini-course, we consider the Ising model on combinatorial maps (more precisely cubic maps, dual to triangulations). Combinatorially, this consists in enumerating these maps, with vertices colored black or white, weighted by their number of monochromatic edges. We want to calculate the associated partition function, once resummed over all maps with a given genus and a given size.In genus 0, the Ising model is combinatorially well understood, with exact formulas available. For greater genus, topological recursion provides an algorithmic way to compute the aforementioned partition functions. However, the computational complexity quickly explodes as the genus grows, which obstructs in particular the study of the high genus regime.
In a joint work with Mireille Bousquet-Mélou, we provide a polynomial time algorithm to compute partition functions in every genus. Our main tools are the classical correspondence between the Ising model and bipartite maps, as well as the KP hierarchy, a system of PDEs that is satisfied by the generating function of combinatorial maps (and many other models of enumerative geometry).In a first part, we will provide some context around the Ising model and maps, and in a second part, we will sketch the outline of the proof of our result.
Exposé invité (1h) d'Aurélien Grabsch : A closed equation for dynamical correlations in the symmetric simple exclusion process
Résumé:
The symmetric simple exclusion process (SEP) is a minimal model yet paradigmatic model of transport in narrow channels. The distributions of different quantities, like the integrated current or the displacement of a tagged particle, have been determined explicitly relying on tools from integrable probabilities. In particular, a tagged particle displays a subdiffusive behaviour, due to the hardcore interactions with the other particles. To quantify this effect, in this talk, we study the correlations between the displacement of a particle and the density of surrounding particle. Although this is a many-body system, we show that these correlations obey at large scales a surprisingly simple closed equation, from which all previous results are easily recovered. Additionally, this equation quantifies the coupling and the spatial response of the system to the displacement of a tagged particle. These results can be extended to other systems and other observables.