Ising Model on Random Triangulations of the Disk: Phase Transition

Area of Science:

• Statistical Mechanics
• Random Matrix Theory
• Quantum Gravity

Background:

• Previous work studied the Boltzmann random triangulation of a disk with an Ising model at critical temperature.
• The Dobrushin boundary condition imposes constraints on the Ising model's spin clusters.

Purpose of the Study:

• To investigate the phase transition of the coupled random triangulation-Ising model at arbitrary temperatures.
• To compute the partition function and derive critical exponents for the infinite perimeter limit.
• To generalize the local limit construction to a diagonal asymptotic regime.

Main Methods:

• Extension of previous results to arbitrary temperatures.
• Computation of the partition function and derivation of critical exponents.
• Analysis of the local limit properties at high and low temperatures.
• Generalization of the local limit construction to a diagonal asymptotic regime.

Main Results:

• The partition function is computed for all temperatures.
• The model exhibits a temperature-dependent local limit, transitioning from percolation-like behavior at high temperatures to a bottleneck structure at low temperatures.
• A novel order parameter is introduced to characterize this temperature-driven change.
• A scaling limit related to the Ising interface length is obtained in the diagonal regime, aligning with Liouville quantum gravity predictions.

Conclusions:

• The phase transition of the Boltzmann random triangulation-Ising model is fully characterized across all temperatures.
• The study introduces a new order parameter with geometric significance.
• The results provide insights into quantum surfaces and Liouville quantum gravity.