Critical Ising Model on Random Triangulations of the Disk: Enumeration and Local Limits

Abstract

In [ ], we have studied the Boltzmann random triangulation of the disk coupled to an Ising model with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the in nite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform in nite half-planar triangulation (UIHPT) decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of nite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning. In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in [ ] to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of quantum surfaces (a.k.a. Liouville quantum gravity). their roots in the physics literature [ ], and was revisited and popularized by Angel in [ ]. The peeling process proves to be a valuable tool for understanding the geometry of random planar maps without Ising model, see [ ] for a review of recent developments.In a previous paper [ ], we extended some enumeration results of Bernardi and Bousquet-Mélou [ ] to study the Ising-decorated random triangulations with Dobrushin boundary condition at its critical temperature. We used the peeling process to construct the local limit of the model, and to obtain several scaling limit results concerning the lengths of some Ising interfaces. In this paper, we extend similar results to the model at any temperature, and show how the large scale geometry of Ising-decorated random triangulations changes qualitatively at the critical temperature. In particular, our results con rm the physical intuition that, at large scale, Ising-decorated random maps at non-critical temperatures behave like non-decorated random maps.A similar model of Ising-decorated triangulations has been studied in a recent work of Albenque, Ménard and Schae er [ ]. They followed an approach reminiscent of Angel and Schramm in [ ] to show that the model has a local limit at any temperature, and obtained several properties of the limit such as one-endedness and recurrence for some temperatures. However they studied the model without boundary, and hence did not encounter the geometric consequences of the phase transition.We start by recalling some essential de nitions from [ ].Planar maps. Recall that a ( nite) planar map is a proper embedding of a nite connected graph into the sphere S 2 , viewed up to o...

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

“…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].…”

“…More recently, the peeling process has also been applied to study the random triangulations of the disk coupled to the Ising model on faces by Chen and the author in the works [8] and [9]. There, the authors have developed a machinery based on analytic combinatorics and rational parametrizations to understand the asymptotic behavior of the partition functions, and constructed local limits using the infinite boundary limits of the perimeter processes associated with the peeling process.…”

Multi-ended Markovian triangulations and robust convergence to the UIPT

“…On the other hand, we recall that C λ,γ (x) = p≥1 C p (λ, γ)x p . By solving (11) and using Z 0 (x) = 0, we get:…”

“…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

“…We simplify it further with Möbius transforms of V to get the parametrization of the theorem. See also [27,Remark 9] for details on this guess and check procedure.…”