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

Erik Duse (KTH Stockholm)

The variational problem for dimer models. Classification of regularity of minimisers and frozen boundaries

In this talk I will introduce the variational problem for dimer models and describe some if its analytical difficulties, in particular the lack of differentiability of dimer functionals and how it connects to solutions of Monge-Ampère equations on planar convex compact polygons. Using a remarkable connection to quasiconformal mappings and Beltrami equations, I will show how these analytic difficulties can be overcome and how one can obtain optimal regularity results for minimisers and frozen boundaries. If time permits, some intriguing open problems connected to the probabilistic theory of dimer models will be explained.

Jeanne Scott

Perturbing Isoradial Triangulations (Slides)

In joint work with François David, we consider an infinite, planar Delaunay graph $\mathcal{G}_\epsilon$ which is obtained by locally perturbing the coordinate embedding of a general, isoradial graph $\mathcal{G}_\mathrm{cr} $ with respect to real-valued, deformation parameter(s) $\epsilon$. Three Laplace-like operators are supported on such a Delaunay graph $\mathcal{G}_\epsilon$, namely: The Beltrami-Laplace operator $\Delta(\epsilon)$, the David-Eynard Kähler operator $\mathcal{D}(\epsilon)$, and the conformal Laplacian $\underline{\Delta}(\epsilon)$. All three operators coincide with Kenyon's critical Laplacian when restricted to the initial, isoradial graph $\mathcal{G}_\mathrm{cr}$ and, as such, each operator can be viewed as a independent deformation of the critical Laplacian. Using Kenyon's asymptotic results for the Green's function of the critical Laplacian, we calculate the leading asymptotics of the first and second order terms in the perturbative expansion of the log-determinant of each operator. The first order term allow us to define a discrete stress energy tensor for each operator. In the case of the Beltrami-Laplace and discrete Kähler operators, the second order term share a convergent scaling limit which is independent of the initial isoradial graph $\mathcal{G}_\epsilon$; for the Beltrami-Laplace operator this limit is consistent with the results of the continuum theory (i.e. the GFF).

Pavel Galashin (UCLA)

Ising model, total positivity, and criticality (Slides)

I will discuss a recent connection between the planar Ising model and the totally nonnegative orthogonal Grassmannian (joint work with Pavlo Pylyavskyy). I will then explain a boundary correlation formula for the critical Ising model and its extension to the critical dimer model inside the totally nonnegative Grassmannian.

Rémy Mahfouf (École Normale Supérieur de Paris)

Universality of spin correlations in the Ising model on isoradial graphs (Slides)

We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs and Z–invariant weights. Specifically, we show that in the massive scaling limit, i. e., as the mesh size tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations converges to a rotationally invariant function, which is universal among isoradial graphs satisfying the bounded angles property and independent of the local geometry. Those convergence results remains true (at criticality) for the 2D graphical representation of the Quantum Ising, which provides explicit formulae for the scaling limit of space-time spin correlations of the 1D Quantum Ising model. We also give a simple proof of the fact that the infinite-volume magnetization in the Z–invariant model is independent of the site and the local geometry of the lattice. Based on joint works with Dmitry Chelkak (ENS), Konstantin Izyurov (Helsinki) and Jhih-Huang Li (Warwick).

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.