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We investigate the geometry of a typical spin cluster in randomtriangulations sampled with a probability proportional to the energy of anIsing configuration on their vertices, both in the finite and infinite volumesettings. This model is known to undergo a combinatorial phase transition at anexplicit critical temperature, for which its partition function has a differentasymptotic behavior than uniform maps. The purpose of this work is to givegeometric evidence of this phase transition.  In the infinite volume setting, called the Infinite Ising PlanarTriangulation, we exhibit a phase transition for the existence of an infinitespin cluster: for critical and supercritical temperatures, the root spincluster is finite almost surely, while it is infinite with positive probabilityfor subcritical temperatures. Remarkably, we are able to obtain an explicitparametric expression for this probability, which allows to prove that thepercolation critical exponent is $\beta=1/4$.  We also derive critical exponents for the tail distribution of the perimeterand of the volume of the root spin cluster, both in the finite and infinitevolume settings. Finally, we establish the scaling limit of the interface ofthe root spin cluster seen as a looptree. In particular in the wholesupercritical temperature regime, we prove that the critical exponents and thelooptree limit are the same as for critical Bernoulli site percolation.  Our proofs mix combinatorial and probabilistic arguments. The starting pointis the gasket decomposition, which makes full use of the spatial Markovproperty of our model. This decomposition enables us to characterize the rootspin cluster as a Boltzmann planar map in the finite volume setting. We thencombine precise combinatorial results obtained through analytic combinatoricsand universal features of Boltzmann maps to establish our results.

Geometric properties of spin clusters in random triangulations coupled with an Ising Model