Dimers: from combinatorics to quantum mechanics – DIMERS
The dimer model has been introduced in statistical mechanics to describe the
adsorption of molecules on the surface of the crystal in chemistry.
Mathematically, it is defined as a probability measure on perfect matchings of a
graph, also called dimer configurations.
The dimer model on a planar graph is linked to several other models of
interest: random tilings, spanning trees, the Ising model, etc.
Several approaches to this model are available, due to its connection to
representation theory, discrete complex analysis and algebraic geometry.
The goal of the DIMERS project is to study several aspects of the dimer models
and other related models from statistical mechanics: limit shapes and
fluctuations, dynamics, integrability and connection with gauge theory.
The objectives are splitted into five interdependent and complementary axes:
A) Limit shapes and fluctuations
This task is devoted to problems related to the deterministic
limit for the normalized height function encoding random dimer configurations,
or the arctic curve separating regions with different local behavior, together
with fluctuations around these limits. These are instances of law of
large numbers and central limit theorem for these geometric random objects.
B) Gauge theory
This task deals with the development of techniques around discrete differential
operators on vector bundles over graphs and their applications to the study of
topological observables for double dimers, spanning forests and related models.
This part focuses on dynamical aspects of these models from different points of
view: mixing time for Glauber dynamics, algorithms for perfect simulations of
random configurations but also the interpretion as 2D static dimer
configurations as the evolution of quantum 1D systems via Wick's rotation.
D) Statistical mechanics and integrability on isoradial graphs
In this task, we investigate dimer models and related Z-invariant models of
statistical mechanics on a special family of embedded planar graphs, in
connection with spectral curves, discrete harmonic analysis and integrability, as well
as s-embeddings of critical Ising models on planar graphs.
E) Models with interactions
This task is about extended existing results on dimers to models with
interactions, which don't have anymore a determinantal structure.
The multidisciplinary consortium gathered around this project is composed of probabilists, combinatorists and
physicists. Various techniques will be combined, from combinatorial bijections,
to quantum field theory, including representation theory, methods of
differential geometry and discrete complex analysis. This multiplicity of
techniques show that theses models stand at the intersection of several branches
of mathematics. We expect that bringing together this diversity of approaches will not only
convey important developments on dimer models but also help techniques and
know-how to percolate between the different communities involved, both within
the consortium and at a larger scale.
Monsieur Cédric Boutillier (Laboratoire de Probabilités, Statistiques et Modélisations)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
LPSM Laboratoire de Probabilités, Statistiques et Modélisations
LP ENSL LABORATOIRE DE PHYSIQUE DE L'ENS DE LYON
Help of the ANR 275,181 euros
Beginning and duration of the scientific project: February 2019 - 48 Months