Local convergence of large random triangulations coupled with an Ising model
| Autor: | Gilles Schaeffer, Laurent Ménard, Marie Albenque |
|---|---|
| Přispěvatelé: | Laboratoire d'informatique de l'École polytechnique [Palaiseau] (LIX), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Université Paris Nanterre (UPN), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Modélisation aléatoire de Paris X (MODAL'X), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014), ANR-16-CE40-0009,GATO,Graphes, Algorithmes et TOpologie(2016), ANR-11-LABX-0023,MME-DII,Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions(2011) |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Computer Science::Machine Learning
Phase transition 82B44 General Mathematics [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] Computer Science::Digital Libraries 01 natural sciences Measure (mathematics) Statistics::Machine Learning 010104 statistics & probability [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] FOS: Mathematics Mathematics - Combinatorics 60C05 Limit (mathematics) Boundary value problem 0101 mathematics 60D05 Mathematics Spins Applied Mathematics 010102 general mathematics Mathematical analysis Probability (math.PR) Mathematics subject classification 05A15 Simple random sample Local convergence [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] 05A16 60K35 Computer Science::Mathematical Software Ising model 05C30 Combinatorics (math.CO) Mathematics - Probability |
| Zdroj: | Transactions of the American Mathematical Society Transactions of the American Mathematical Society, 2021, 374 (1), pp.175-217. ⟨10.1090/tran/8150⟩ Transactions of the American Mathematical Society, American Mathematical Society, 2021, 374 (1), pp.175-217. ⟨10.1090/tran/8150⟩ |
| ISSN: | 0002-9947 1088-6850 |
| DOI: | 10.1090/tran/8150⟩ |
| Popis: | We prove the existence of the local weak limit of the measure obtained by sampling random triangulations of size $n$ decorated by an Ising configuration with a weight proportional to the energy of this configuration. To do so, we establish the algebraicity and the asymptotic behaviour of the partition functions of triangulations with spins for any boundary condition. In particular, we show that these partition functions all have the same phase transition at the same critical temperature. Some properties of the limiting object -- called the Infinite Ising Planar Triangulation -- are derived, including the recurrence of the simple random walk at the critical temperature. 42 pages, 9 figures |
| Databáze: | OpenAIRE |
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