Local convergence of large random triangulations coupled with an Ising model

Albenque, Marie; Ménard, Laurent; Schaeffer, Gilles

doi:10.1090/tran/8150

Local convergence of large random triangulations coupled with an Ising model
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by Marie Albenque, Laurent Ménard and Gilles Schaeffer PDF

Trans. Amer. Math. Soc. 374 (2021), 175-217 Request permission

Abstract:

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.

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