**Abstract:** I will prove that the Ising model on any nonamenable Cayley graph undergoes a continuous phase transition. I will also review some relevant techniques from Bernoulli percolation, most notably those of https://arxiv.org/pdf/1808.08940.pdf: A key step of the proof is to adapt the methods of that paper from Bernoulli percolation to certain ‘percolation in random environment’ models that do not necessarily satisfy FKG, which we apply to the (double) random current representation of the Ising model. The proof relies crucially on the spectral theory of group-invariant stochastic processes, which I will also review.

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