Les journées de l’ANR DIMERS ont eu lieu à Paris, les 8 et 9 octobre 2019 à Sorbonne Université, campus Pierre et Marie Curie (métro Jussieu).
Le café d’accueil et les exposés du mardi matin ont eu lieu en salle 16-26-113 (premier étage du couloir entre les tours 16 et 26).
Les exposés du mardi après-midi ont eu lieu en salle 16-26-209 (salle Paul Lévy, deuxième étage du couloir entre les tours 16 et 26).
Toutes les activités de l’ensemble de la journée du mercredi ont eu lieu en salle 15-16-201 (deuxième étage du couloir entre les tours 15 et 16).
Matinée en salle 16-26-113 :
Après-midi en salle 16-26-209 :
Journée complète en salle 15-16-201:
In discrete differential geometry (DDG) we study discretizations of maps and surfaces that are structure preserving. In recent research dimer systems have appeared in several objects of DDG. We will focus on three examples where dimers appear: Circle patterns are a notion of discrete conformal maps and dimers appear while studying Miquel dynamics, a dynamical system on the space of circle patterns. As a second example we look at surfaces that have been parametrized as conjugate nets. The corresponding discretization is Q-Nets, these are maps from Z^2 to R^3 such that each face is planar. We show how dimer weights can be defined on Q-Nets such that the partition function is invariant with respect to the so called Laplace transforms. Finally, we look at the discrete Schwarzian KP (dSKP) equation. The discrete KP equation, also known as the octahedron recurrence appears both in Dodgson condensation and the Aztec Diamond Theorem. The dSKP equation plays a key role in the integrability properties of DDG. We show there is an analogue to Dodgson condensation in the dSKP case and finish with an open question: Is there a Schwarzian Aztec Diamond Theorem?
I will discuss how to obtain explicitly the arctic curves of a number of specific non-intersecting or osculating path models by the somewhat simple technique of the tangent method. The models include non-intersecting paths with an arbitrary distribution of starting points as well as osculating path configurations describing the so-called 20-vertex model (the ice model on the triangular lattice) with particular domain wall boundary conditions. This is a joint work with P. Di Francesco and B. Debin.
Basing upon a joint work in progress with Benoît Laslier and Marianna Russkikh, we discuss a concept of `perfect t-embeddings’ of weighted bipartite planar graphs onto the unit disc D. (T-embeddings also appeared under the name Coulomb gauges in a recent work of Kenyon, Lam, Ramassamy and Russkikh.) Though the overall picture is still incomplete (in particular we do not know that such embeddings always exist), we believe that, in many setups, they are good candidates to recover the complex structure of big planar graphs carrying a bipartite dimer model.
In particular, given a sequence of (abstract) planar graphs G_n and their perfect t-embeddings T_n onto D, assume that (i) the faces of T_n satisfy certain technical assumptions in the bulk of D; (ii) on each compact subset of D, the size of the associated origami maps O_n tends to zero as n grows. We prove that (i)+(ii) imply the convergence of the fluctuations of the dimer height functions on G_n (provided that these graphs are embedded by T_n), to the GFF on the unit disc D equipped with the standard complex structure.
Moreover, the same techniques seem to work in the situation when the limit of origami maps T_n->O_n arising from a sequence of perfect t-embeddings is a minimal surface in the Lorentz metric (and we optimistically conjecture that this is always the case). In this situation one should eventually obtain the GFF in the Enneper-type parameterization of the origami profile.
Schur processes form a general class of models in integrable probability. Their applications range from dimer models (plane partitions, domino tilings of the Aztec diamond…) to last-passage percolation (LPP). They display a rich asymptotic behaviour, some of it belonging to the “KPZ universality class”.
I will discuss Schur processes with periodic and free boundary conditions. The periodic case can be understood in terms of free fermions at finite temperature, while the free boundary case involves “superconducting” fermionic states. I will discuss applications (cylindric partitions, LPP) and the appearance of new asymptotic behaviours.
This talk is based on joint work with Dan Betea, Peter Nejjar and Mirjana Vuletić.
Penrose tilings are aperiodic rhombus tilings often used to model quasicrystals. They are known to be characterized by a finite set of forbidden patterns (which aim to model finite-range energetic interactions). What happens if we add to the model a temperature parameter which allows forbidden patterns to appear (the lower the temperature is - the less forbidden patterns are allowed) ? More precisely, does exist a phase transition, that is, a critic temperature below which typical tilings stays “close” to Penrose tilings while they go wild above this temperature? In this talk we shall review two papers published in the same volume of Physical Review B (vol 41, 1990) which support opposite views on this problem. We also want to generalize the question to any rhombus tilings (including the dimer tilings the audience is more familiar with) and discuss the role played by so-called flip dynamics in this issue.