Abstract

In this talk, I consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point. First, the partition function is computed and the perimeter exponent shown to be 7/3 instead of the exponent 5/2 for uniform triangulations. Then, I sketch the  construction of the local limit in distribution when the two components of the Dobrushin boundary tend to infinity one after the other, using the peeling process along an Ising interface. In particular, the main interface in the local limit touches the (infinite) boundary almost surely only finitely many times, a behavior opposite to that of the Bernoulli percolation on uniform maps. Some scaling limits closely related to the perimeters of clusters are also discussed. This is based on a joint work with Linxiao Chen.