Interfaces in the vertex-decorated Ising model on random triangulations of the disk

Abstract

We provide a framework to study the interfaces imposed by Dobrushin boundary conditions on the half-plane version of the Ising model on random triangulations with spins on vertices. Using the combinatorial solution by Albenque, Ménard and Schaeffer ([2]) and the generating function methods introduced by Chen and Turunen ([8], [9]), we show the local weak convergence of such triangulations of the disk as the perimeter tends to infinity, and study the interface imposed by the Dobrushin boundary condition. As a consequence of this analysis, we verify the heuristics of physics literature that discrete interface of the model in the high-temperature regime resembles the critical site percolation interface, as well as provide an explicit scaling limit of the interface length at the critical temperature, which coincides with results on the continuum Liouville Quantum gravity surfaces. Overall, this model exhibits simpler structure than the model with spins on faces, as well as demonstrates the robustness of the methods developed in [8], [9].

arXiv comment35 pages, 8 figures, 4 tables. Minor modifications

References in corpus (6)

• Scaling of Percolation on Infinite Planar Maps, I
• The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen)
• Liouville Quantum Gravity on the Riemann sphere
• Ising model on random triangulations of the disk: phase transition
• Growth and Percolation on the Uniform Infinite Planar Triangulation
• Markov chains with heavy-tailed increments and asymptotically zero drift

Cited by in corpus (1)

• Ising model on random triangulations of the disk: phase transition