# Local convergence of large random triangulations coupled with an Ising model

## TL;DR

This paper establishes the local weak limit of large random triangulations decorated with an Ising model, analyzing their partition functions and properties of the resulting infinite triangulation at criticality.

## Contribution

It proves the existence of the local limit for random triangulations with Ising configurations and characterizes their phase transition and properties at critical temperature.

## Key findings

- Partition functions are algebraic and share the same phase transition.
- The Infinite Ising Planar Triangulation is recurrent for simple random walk at criticality.
- The critical temperature is the same for all boundary conditions.

## 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.