**Organizers:**- Cédric Boutillier (Sorbonne Université)
- Dmitry Chelkak (École Normale Supérieure)

Adrien Kassel (ENS Lyon)

Discrete vector bundles and determinantal processes (Slides)

I will report on joint work with Thierry Lévy in which we attempt to give probabilistic meaning to the laplacian determinant on a discrete vector bundle of arbitrary rank.

Helen Jenne (Tours)

Double-dimer condensation and the dP3 Quiver (Slides)

In the first half of this talk we will discuss a new result about the double-dimer model: under certain conditions, the partition function for double-dimer configurations of a planar bipartite graph satisfies a recurrence related to Dodgson condensation (also called the Desnanot-Jacobi identity). A similar identity for the number of dimer configurations of a planar bipartite graph was established nearly 20 years ago by Kuo. In the second half of the talk, we will describe an application of this condensation result to a problem in cluster algebras, which is ongoing joint work with Tri Lai and Gregg Musiker. In 2017, Lai and Musiker gave combinatorial interpretations for many toric cluster variables in the cluster algebra associated to the cone over the del Pezzo surface dP3. Specifically, they used Kuo condensation to show that most toric cluster variables have Laurent expansions agreeing with partition functions for dimer configurations. However, in some cases, the dimer model was not sufficient. We show that in these cases, the Laurent expansions agree with partition functions for double-dimer configurations.

Sunil Chhita (Durham)

The domino shuffling algorithm and anisotropic KPZ stochastic growth

The domino-shuffling algorithm can be seen as a stochastic process describing the irreversible growth of a (2+1)-dimensional discrete interface. Its stationary speed of growth depends on the average interface slope, as well as on the edge weights, that are assumed to be periodic in space. In this talk, we present the model and also some results showing that this model belongs to the anisotropic KPZ class for general periodic weights. Finding an explicit formula for the speed of growth is out of reach. Instead we show a relation between the speed of growth and the limiting height function of domino tilings of the Aztec diamond. This is joint work with Fabio Toninelli.

David Cimasoni (Genève)

Dimers on the Klein bottle

The dimer model on graphs embedded in the torus (or equivalently, doubly-periodic planar graphs) is very well understood thanks to the classical work of Kasteleyn, Kenyon, Okounkov, Sheffield, and many others. In this talk, I will report on ongoing work about the dimer model on graphs embedded in the Klein bottle (or equivalently, periodic-antiperiodic planar graphs). As we will see, Kasteleyn's theorem extends (this is due to Tesler), as well as Cohn-Kenyon-Propp's double product formula (joint work with Adrien Kassel). This allows us to evaluate the asymptotics of the dimer partition function on Klein bottles, in the spirit of Kenyon, Sun, and Wilson's result on tori.