Modèle de forêts enracinées sur des cycles et modèle de perles via les dimères

• Main Author: Sun, Wangru
• Other Authors: Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université, Béatrice de Tilière, Cédric Boutillier
• Format: Doctoral or Postdoctoral Thesis
• Language: English
• Published: HAL CCSD 2018
• Subjects: Dimer model; Tilings; Cycle-rooted-spanning-forests; Random walks; Bead model; Modèles de dimères; Pavages; Forêts couvrantes enracinées sur cycles; Marches aléatoires; Modèle de perles; Tableaux de Young standard; [MATH.MATH-PR]Mathematics [math]/Probability [math.PR]; Arctic; Random Point
• Online Access: https://tel.archives-ouvertes.fr/tel-01734698; https://tel.archives-ouvertes.fr/tel-01734698v2/document; https://tel.archives-ouvertes.fr/tel-01734698v2/file/2018SORUS007.pdf

The dimer model, also known as the perfect matching model, is a probabilistic model originally introduced in statistical mechanics. A dimer configuration of a graph is a subset of the edges such that every vertex is incident to exactly one edge of the subset. A weight is assigned to every edge, and the probability of a configuration is proportional to the product of the weights of the edges present. In this thesis we mainly study two related models and in particular their limiting behavior. The first one is the model of cycle-rooted-spanning-forests (CRSF) on tori, which is in bijection with toroidal dimer configurations via Temperley's bijection. This gives rise to a measure on CRSF. In the limit that the size of torus tends to infinity, the CRSF measure tends to an ergodic Gibbs measure on the whole plane. We study the connectivity property of the limiting object, prove that it is determined by the average height change of the limiting ergodic Gibbs measure and give a phase diagram. The second one is the bead model, a random point field on Z x R which can be viewed as a scaling limit of dimer model on a hexagon lattice. We formulate and prove a variational principle similar to that of the dimer model [CKP01], which states that in the scaling limit, the normalized height function of a uniformly chosen random bead configuration lies in an arbitrarily small neighborhood of a surface h₀ that maximizes some functional which we call as entropy. We also prove that the limit shape h₀ is a scaling limit of the limit shapes of a properly chosen sequence of dimer models. There is a map form bead configurations to standard tableaux of a (skew) Young diagram, and the map is measure preserving if both sides take uniform measures. The variational principle of the bead model yields the existence of the limit shape of a random standard Young tableau, which generalizes the result of [PR07]. We derive also the existence of an arctic curve of a discrete point process that encodes the standard tableaux, raised in [Rom12]. Le ...