The sharp factor-of-IID threshold for the free Ising model on regular trees
For every integer d ≥ 3 and inverse temperature β ≥ 0, the free zero-field ferromagnetic Ising measure on the infinite d -regular tree is a factor of IID exactly when $\tanh\beta\le(d-1)^{-1/2}$. We prove the positive implication, including equality, resolving the conjecture of Nam, Sly and Zhang; the strict converse is known. The spin factor can be chosen to commute with every tree automorphism on every label input.
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