La journée de rentrée l’ANR DIMERS a eu lieu à Paris, le 28 septembre 2021 à Sorbonne Université, campus Pierre et Marie Curie (métro Jussieu).
Cette journée a comporté principalement des présentations de doctorants et post-docs travaillant dans le périmètre élargi du projet.
Les exposés ont eu lieu dans la salle Paul Lévy, 16-26-209 (2e étage du couloir entre les tours 16 et 26) et ont été retransmis en ligne.
Tau-functions à la Dubédat and cylindrical events in the double-dimer model /slides/
We discuss the convergence of the double-dimer random loop ensembles, sampled in a sequence of discrete Temperleyn domains approximating a given simply-connected domain on the plane. It was conjectured by R. Kenyon that, in this setup, these double-dimer loop ensembles converge to the conformal loop ensemble with parameter 4 (CLE(4)) in the given domain. Based on the recent work of Julien Dubédat, we prove that probabilities cylindrical events for these double-dimer loop ensembles converge, and we describe their limits in terms of the Jimbo-Miwa-Ueno isomonodromic tau function. Moreover, using crossing estimates for CLE, recently obtained by Tianyi Bai and Yijun Wan, and the result of Dubédat, that express the topological correlators of CLE(4) via isomonodromic tau functions, we are able to identify the limits of these probabilities with probabilities taken with respect to CLE(4). Since the probabilities of cylindrical events altogether characterize the measure, this implies that if the double-dimer loop ensembles that we consider converge to any random loop ensemble, then this limit ensemble must coincide with CLE(4). Based on a joint work with Dmitry Chelkak.
Two-dimensional Markov Processes, a higher algebraic approach
When working with one-dimensional Markov Processes, right and left eigenvectors of the transfer matrix provide agood notion of invariant boundary conditions. Unfortunately, from what we can tell, such a tool doesn’t exist for two-dimensional Markov Processes. Recent work by Damien Simon fills this gap using a higher algebraic approach based on the theory of Operads. In this talk we’ll present the algebraic objects that arise when considering this new formalism and how they define invariant boundary conditions.
Quantum spanning forests and phase transition /slides/
I will talk about the statistical physics model of so-called quantum spanning forests, introduced by A. Kassel and T. Lévy, which generalizes the classical model of the uniform spanning tree on a graph. This model is a determinantal process, which depends on a choice of unitary connection on a vector bundle over the graph. I will consider the case of a periodic connection on a d-periodic graph, which can be approximated by graphs on a sequence of growing tori, and I will explicit the kernel of the process in that case. This allows to identify two phases, based on the rate of decay of correlations, which depend on properties of the connection. I will finally talk about the particular case of a rank 2 vector bundle and its link with quaternionic connections.
An autumn stroll among random tilings (at low temperatures with local rules) /slides/
In the context of statistical physics, quasicrystals are highly-structured yet aperiodic materials. The Penrose tiling, with its fivefold rotational invariance, is a good candidate to study their low-temperature stability. As a first stepping stone, we see here that a random dimer model with local interactions, seen as a random surface though the height function, stays close to the zero-temperature surface on average.
In the context of information theory, we are interested in the robustness of computations to a low amount of random mistakes. Most particularly, through the lense of natural computing, the Robinson tiling is a hierarchical structure that can encode such computations. We see here that the Robinson structure is stable in the sense that in a generic noisy tiling, a high proportion of tiles match the same ground configuration.
The goal of this talk is to see how these two viewpoints compare with each other, in particular regarding stability when the amount of mistakes goes to 0.
Boundary Correlations for the Z-invariant Ising Model /notes/
Using dimers, a recent work by P. Galashin and P. Pylyavskyy establishes a link between boundary correlation matrices of planar Ising models embedded in a disk and matrices in a set called the positive orthogonal Grassmannian. In this talk, I will explain this link as well as its practical use to compute correlations for the Z-invariant Ising model.
Geometric shapes from discrete holomorphic functions /slides/
Minimal surfaces are an important class of surfaces in differential geometry. They are characterized as those surfaces that minimize local area and have been studied since the 18th century. Examples of these surfaces proved difficult to find until Wierstrass and Enneper introduced a representation formula: given a holomorphic function, the Weierstrass-Enneper representation allows to construct a minimal surface and all minimal surfaces arise this way.
In the more recently developed field of discrete differential geometrty, discrete minimal surfaces were one of the first object classes to be discretized. In 1996, Bobenko and Pinkall gave a geometric definition of discrete minimal surfaces and proved the existence of a discrete Weierstrass-Enneper representation.
In this talk, we will introduce discrete differential geometry and a notion of discrete mean curvature for discrete isothermic surfaces. We will discuss the dual transformation of this class of surfaces and show that the discrete Weierstrass-Enneper representation constitutes a special instance of this transformation.
Ising model on random triangulations with a boundary /slides/
We consider random planar triangulations of the disk coupled with an Ising model (either on the faces or the on the vertices) at a fixed temperature with Dobrushin boundary conditions, emphasizing their local limits as the perimeter of the disk tends to infinity. We identify rigorously a phase transition by analysing the critical behaviour of the partition functions at and around the critical point. Moreover, we study the geometric implications of this in particular to the interfaces between the spin clusters. At the critical temperature, we find some explicit scaling limits of observables related to the infinite interface which have interpretations in the continuum Liouville Quantum Gravity coupled with matter. The two key techniques in use are singularity analysis of rational parametrizations together with analytic combinatorics, as well as an exploration process of the Ising interface. Based on joint work with Linxiao Chen.