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We investigate the geometry of a typical spin cluster in random\ntriangulations sampled with a probability proportional to the energy of an\nIsing configuration on their vertices, both in the finite and infinite volume\nsettings. This model is known to undergo a combinatorial phase transition at an\nexplicit critical temperature, for which its partition function has a different\nasymptotic behavior than uniform maps. The purpose of this work is to give\ngeometric evidence of this phase transition.\n In the infinite volume setting, called the Infinite Ising Planar\nTriangulation, we exhibit a phase transition for the existence of an infinite\nspin cluster: for critical and supercritical temperatures, the root spin\ncluster is finite almost surely, while it is infinite with positive probability\nfor subcritical temperatures. Remarkably, we are able to obtain an explicit\nparametric expression for this probability, which allows to prove that the\npercolation critical exponent is $\\beta=1/4$.\n We also derive critical exponents for the tail distribution of the perimeter\nand of the volume of the root spin cluster, both in the finite and infinite\nvolume settings. Finally, we establish the scaling limit of the interface of\nthe root spin cluster seen as a looptree. In particular in the whole\nsupercritical temperature regime, we prove that the critical exponents and the\nlooptree limit are the same as for critical Bernoulli site percolation.\n Our proofs mix combinatorial and probabilistic arguments. The starting point\nis the gasket decomposition, which makes full use of the spatial Markov\nproperty of our model. This decomposition enables us to characterize the root\nspin cluster as a Boltzmann planar map in the finite volume setting. We then\ncombine precise combinatorial results obtained through analytic combinatorics\nand universal features of Boltzmann maps to establish our results.\n