La cinquième rencontre de l’ANR Dimers a eu lieu à l’Institut Camille Jordan (Lyon 1, campus de la Doua à Villeurbanne) les mardi 29 et mercredi 30 mars 2022. Organisateurs : Jérémie Bouttier, Adrien Kassel, Jean-Marie Stéphan.

Accès depuis la gare Lyon Part-Dieu : prendre le tram T1 direction IUT Feyssine ou bien le T4 direction La Doua, et sortir à l’arrêt Université Lyon 1. En sortant du tram, le bâtiment Braconnier se situe sur la gauche, dans le sens de la marche. Les salles 112 et Fokko du Cloux sont au premier étage, en sortant sur la droite de l’un des escaliers. La salle Fontannes (mardi après-midi) se situe au rez-de-chaussée du bâtiment Darwin D à 10 mètres de l’arrêt de tram, sur la droite dans le sens de la marche.

Matinée en salle 112, Bâtiment Braconnier, 1er étage.

**9h30-10h30 : petit-déjeuner d’accueil**(café, thé, jus de fruits, brioches pralines et chocolat, cookies, fruits frais)**10h30-10h40 : Hommage à Krzysztof Gawedzki****10h40-11h30 : Sanjay Ramassamy**(Paris-Saclay)**11h30-12h20 : Matteo Mucciconi**(Warwick)**12h20-14h20 : déjeuner au restaurant Domus**

Après-midi en salles Fontannes, du Bâtiment Darwin D, rez-de-chaussée.

**14h20-15h10 : Yijun Wan**(ENS-Paris)**15h10-16h : Siarhei Finski**(Polytechnique)**16h-17h : pause café**

Salle 112, Bâtiment Braconnier, 1er étage.

**17h-18h :**Discussion sur la gestion administrative et scientifique du projet DIMERS**19h : dîner au restaurant**

Matinée en salle Fokko du Cloux, Bâtiment Braconnier, 1er étage.

**9h30-10h30 : petit-déjeuner d’accueil****10h30-11h20 : Tony Jin**(Genève)**11h20-12h10 : Cédric Boutillier**(Sorbonne Université)**12h10-13h30 : déjeuner libre**

**Sanjay Ramassamy: Integrable dynamics on polygons and the dimer integrable system**

On the one hand, several discrete-time dynamical systems on spaces of polygons have been shown in the last twenty years to be integrable. On the other hand, Goncharov and Kenyon introduced ten years ago an integrable system associated with the dimer model on bipartite graphs on the torus. Building upon the notion of triple crossing diagram maps (introduced in recent works of Affolter, Glick, Pylyavskyy and myself), I will describe a framework which encompasses both the geometric dynamics on polygons and the dimer integrable system. This framework makes it possible in particular to identify the conserved quantities of both systems. I will illustrate this paradigm on two examples of geometric dynamics : the pentagram map and cross-ratio dynamics.

This talk is based on joint work with Niklas Affolter (TU Berlin and École normale supérieure) and Terrence George (University of Michigan).

**Matteo Mucciconi: Greene invariants for last passage percolation on the cylinder**

In the study of the last passage percolation on the plane, the Greene invariants count the maximal passage time along a fixed number of non intersecting paths. In this talk I will consider the last passage percolation on an infinite cylinder and I will adapt to this setting the notion of Greene invariants. The main result I will present is an explicit description of their probability distribution which relates to Macdonald polynomials. In the proof combination of various techniques will be used, including the theory of Kashiwara crystals and the introduction of a new integrable dynamics based on the celebrated Viennot shadow line construction.

This is based on a joint work with Takashi Imamura and Tomohiro Sasamoto.

**Yijun Wan: On the crossing estimates for simple conformal loops ensembles and the convergence of cylindrical events in the double-dimer model.**

Conformal loop ensembles (CLE) serve as candidates for the scaling limits of certain statistical physics models at critical temperature, which can be interpreted as random collections of disjoint, non-self-crossing loops. For such limiting continuum objects, crossing-type estimates or regularity properties can be instrumental to study the scaling limits of certain models. For simple CLEs, we prove the super-exponential decay of probabilities that there exist n crossings of a quadrilateral uniformly on its conformal mudulus as n goes to infinity. Besides being of independent interest, this also implies the convergence of probabilities of cylindrical events for the double-dimer loop ensembles to those for the nested CLE4 based on the convergence of topological correlators studied by Basok and Chelkak.

This talk is based on a joint work with Tianyi Bai (NYU, Shanghai).

**Siarhei Finski: Spanning trees, graph approximations and zeta-regularized determinant of Laplacian**

By relying on the methods of spectral geometry, we will study the asymptotic of several combinatorial invariants defined on the graph discretizations of flat surfaces, as the mesh of the discretisation tends to zero, and we will see how the geometry of the surface is reflected in this asymptotic.

In particular, we relate the asymptotic expansion of the number of spanning trees on discretizations with the zeta-regularized determinant of the Laplacian, formally defined as the product of all the eigenvalues. Using similar ideas, we deduce conformal invariance of certain observables of loop measures, defined over Riemann surfaces.

**Tony Jin: Quantum symmetric simple exclusion process**

Non-equilibrium phenomena appear in every area of physics. Despite their ubiquity, formulating general principles to describe them remains one of the most important challenges in modern statistical physics.

In the recent decades, much progress has been made in classical statistical physics of driven diffusive system. The combined effort of providing systematic exact solutions to toy models and attempting to formulate general principles that govern them has culminated in the formulation of an unified framework called the macroscopic fluctuation theory.

For the quantum case, the situation is much less advanced, hence the need for simple, solvable, yet non trivial models that will help us grasp the general theory. I will present such model called the quantum symmetric simple exclusion process. After introducing it, I will show that the correlation functions in the stationary state follow a given set of combinatorial rules. From these rules I will demonstrate how one can compute the correlations recursively. The full distribution encodes for a rich behavior, entailing fluctuating quantum coherences which survive in the steady limit and satisfy a large-deviation principle.

**Cédric Boutillier: Minimal bipartite dimers and maximal Riemann surfaces**

A celebrated result on dimer models in the spectral theorem by Kenyon and Okounkov giving a bijection between dimer models on periodic graph modulo some transformations and the space of Harnack curves with a given Newton polygon with a standard divisor (a point on each oval of the curve).

Fock then provided an explicit construction of the inverse spectral map: given an algebraic curve C (not necessarily Harnack) and an appropriate divisor, he defines a periodic minimal graph with a Kasteleyn operator (not necessarily having a combinatorical interpretation) for which C is the spectral curve.

In a joint work David Cimasoni and Béatrice de Tilière, we study the dimer model on arbitrary infinite minimal planar graphs, with Fock’s weights, constructed from a fixed compact Riemann surface. The corresponding Kasteleyn operator has a whole family of inverses with an explicit integral representation, with a certain locality property. We define the notion of divisor for a vertex of the graph. We then explain how in the periodic case, we can obtain a parametrization of the spectral curve, compute the phase diagram, the slope and the free energy directly from the Riemann surface.

These results can be seen as a generalization of Kenyon’s results about isoradial dimers with (genus 0, trigonometric) critical weights to a larger family of graphs, in arbitrary genus.