In [arXiv:1806.06668], we have studied the Boltzmann random triangulation of
the disk coupled to an Ising model on its faces 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
infinite 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 infinite
half-planar triangulation (UIHPT) decorated with a subcritical percolation. At
low temperatures, the local limit develops a bottleneck of finite 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 [arXiv:1806.06668] 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).

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