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].

Ising Model on Random Triangulations of the Disk: Phase Transition

“…However, an analog of the conjecture could be proven for the model with spins on vertices, or with spins on faces and a general boundary. The former is conducted in the preprint [37]. The conjecture is also supported by a prediction derived from the Liouville Quantum Gravity, seen as a continuum model of quantum surfaces studied eg.…”

“…According to Remark 20, there exists an open set V containing H 0 such that Ž is holomorphic in K × V. As Ȟc ∈ V, this implies in particular that Ž is analytic at (H * , Ȟc ). Since (h * ) = H * and (0) = Ȟc , and we have seen that is analytic at both h * and 0, the relation (37) implies that Z is analytic at (h * , 0). It follows that…”

“…In the recent preprint [1], the first two of the aforementioned authors proved several exact results on the perimeter and volume of the spin clusters, demonstrating the phase transition through several critical exponents and geometric behaviors of the cluster in different phases. The model with spins on the vertices can also be studied with a boundary, and the methods introduced in [20] and this article were recently applied to that model in [37] by the second author of this work.…”

Multi-ended Markovian triangulations and robust convergence to the UIPT

“…We note that the case where the Ising model lives on the vertices and the inverse temperature β > 0 is fixed was treated in [1]. We also refer to [11,12,23] for local convergence results on Ising triangulations with a boundary, once again for β fixed.…”

Local convergence of large random triangulations coupled with an Ising model

“…The Ising model was initially introduced by Lenz and studied by Ising in the 1920s [41] to model magnetism, and was proved to exhibit a phase transition in 2-dimension by Onsager [69]. Probabilistic aspects of planar maps decorated by an Ising model were studied recently by the authors and Schaeffer [2], and by Chen and Turunen [26,27], as well as Turunen [73]. In particular, it was proved in [2] that large random planar triangulations coupled with an Ising model converge in law for the local topology for any value of the temperature parameter.…”