Critical Ising model on random triangulations of the disk: enumeration and local limits

TL;DR

This paper analyzes the critical Ising model on random triangulations with Dobrushin boundary conditions, explicitly computes the partition function, and establishes local limits and interface behaviors, revealing new geometric and probabilistic properties.

Contribution

It extends the enumeration of Boltzmann triangulations with Ising models to Dobrushin boundary conditions and constructs explicit local limits using peeling processes.

Findings

Perimeter exponent is 7/3, different from 5/2 in uniform triangulations.

The model exhibits a local limit with a peeling process along an Ising interface.

The main interface touches the boundary finitely many times, unlike percolation on uniform maps.

Abstract

We consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point of the model. The first part of this paper computes explicitly the partition function of this model by solving its Tutte's equation, extending a previous result by Bernardi and Bousquet-M\'elou to the model with Dobrushin boundary conditions. We show that the perimeter exponent of the model is 7/3 in contrast to the exponent 5/2 for uniform triangulations. In the second part, we show that the model has a local limit in distribution when the two components of the Dobrushin boundary tend to infinity one after the other. The local limit is constructed explicitly using the peeling process along an Ising interface. Moreover, we show that 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 finite clusters are also obtained.