Geometric properties of spin clusters in random triangulations coupled with an Ising Model

Understand

We investigate the geometry of a typical spin cluster in random triangulations sampled with a probability proportional to the energy of an Ising configuration on their vertices, both in the finite and infinite volume settings.

• This model is known to undergo a combinatorial phase transition at an explicit critical temperature, for which its partition function has a different asymptotic behavior than uniform maps.
• The purpose of this work is to give geometric evidence of this phase transition.
• In the infinite volume setting, called the Infinite Ising Planar Triangulation, we exhibit a phase transition for the existence of an infinite spin cluster: for critical and supercritical temperatures, the root spin cluster is finite almost surely, while it is infinite with positive probability for subcritical temperatures.