Après avoir expliqué la notion de Z-invariance pour les modèles de mécanique statistique, nous introduisons une famille à un paramètre (dépendant du module elliptique) de Laplaciens massiques Z-invariants définis sur les graphes isoradiaux. Nous démontrons une formule explicite pour son inverse, la fonction de Green massique, qui a la propriété remarquable de ne dépendre que de la géométrie locale du graphe. Nous expliquerons les conséquences de ce résultat pour le modèle des forêts couvrantes, en particulier la preuve d'une transition de phase d'ordre 2 avec le modèle des arbre couvrants critiques sur les graphes isoradiaux, introduit par Kenyon. Finalement, nous considérons la courbe spectrale de ce Laplacien massique et montrons qu'il s'agit d'une courbe de Harnack de genre 1.
Il s'agit d'un travail en collaboration avec Cédric Boutillier et Kilian Raschel.[-]

There is the same number of $n \times n$ alternating sign matrices (ASMs) as there is of descending plane partitions (DPPs) with parts no greater than $n$, but finding an explicit bijection is, despite many efforts, an open problem for about $40$ years now. So far, four pairs of statistics that have the same joint distribution have been identified. We introduce extensions of ASMs and of DPPs along with $n+3$ pairs of statistics that have the same joint distribution. The ASM-DPP equinumerosity is obtained as an easy consequence by considering the $(-1)$enumerations of these extended objects with respect to one pair of the $n+3$ pairs of statistics. One important tool of our proof is a multivariate generalization of the operator formula for the number of monotone triangles with prescribed bottom row that generalizes Schur functions. Joint work with Florian Aigner.[-]

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]

In the framework of the study of K-theoretical Coulomb branches, Finkelberg-Tsymbaliuk introduced remarkable new algebras, the shifted quantum affine algebras an their truncations, in the spirit of the shifted Yangians of Brundan-Kleshev, Braverman-Finkelberg Nakajima, Kamnitzer-Webster-Weekes-Yacobi... We discuss the following points in representation theory of (truncated) shifted quantum affine algebras that we relate to representations of Borel quantum affine algebras by induction and restriction functors. We establish that the Grothendieck ring of the category of finite-dimensional representations has a natural cluster algebra structure. We propose a conjectural parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual quantum affine Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations proved by using the Baxter polynomiality of quantum integrable models.[-]

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]