OAI.Problem367.free_ising_factor_iff_threshold

The theorem states that, for every natural number d ≥ 3 and every real β ≥ 0, a certain property of θ = tanh β holds if and only if tanh β ≤ 1/√(d−1). The property, IsFactorOfIID(d, θ), concerns the d-regular tree whose vertices are the reduced words over the alphabet {0,…,d−1} (finite words in which no two consecutive letters are equal), with two words adjacent when one is obtained from the other by appending a single letter. It asserts that there exists a measurable map Φ from i.i.d. labels in [0,1], one for each tree vertex under the product of uniform measures, to ±1 spin configurations on the vertices, satisfying two conditions. First, the pushforward law of Φ is the free Ising law with parameter θ: for every finite connected vertex set D and every spin assignment η on D, the probability that the spins on D equal η is (1/2)·∏ over edges {u,v} of D of (1+θ·s)/2, where s = +1 if η(u)=η(v) and s = −1 otherwise. Second, Φ is equivariant under every automorphism g of the tree: for almost every label configuration U, applying Φ to the relabelled configuration gives the same result as relabelling Φ(U) by g.