Local convergence of large random triangulations coupled with an Ising\n model

2018/12/07 by Marie Albenque, Albenque, Marie, Laurent Ménard +3 · 5 citations
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1812.03140

openalex publication_date 2018/12/07 · openalex created_date 2020/10/29 · openalex updated_date 2026/08/01

Abstract

We prove the existence of the local weak limit of the measure obtained by\nsampling random triangulations of size n decorated by an Ising configuration\nwith a weight proportional to the energy of this configuration. To do so, we\nestablish the algebraicity and the asymptotic behaviour of the partition\nfunctions of triangulations with spins for any boundary condition. In\nparticular, we show that these partition functions all have the same phase\ntransition at the same critical temperature. Some properties of the limiting\nobject -- called the Infinite Ising Planar Triangulation -- are derived,\nincluding the recurrence of the simple random walk at the critical temperature.\n

Citations

• Scaling of Percolation on Infinite Planar Maps, I
• The degree distribution in bipartite planar maps: applications to the Ising model
• The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen)
• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• Classification of scaling limits of uniform quadrangulations with a\n boundary
• Liouville quantum gravity and the Brownian map III: the conformal\n structure is determined

Cited by

• Ising model with external magnetic field on random planar maps: Critical exponents
• Blossoming bijection for bipartite maps: a new approach via orientations and applications to the Ising model
• The 3-state Potts model on planar triangulations: explicit algebraic solution
• The slice decomposition of planar hypermaps
• Local limit of Prim's algorithm

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• Phase transition for the Ising model on the Critical Lorentzian triangulation
• Critical behavior of the annealed ising model on random regular graphs
• Ising model with external magnetic field on random planar maps: Critical exponents
• The torus plateau for the high-dimensional Ising model