The exact factor-of-IID threshold for free Ising spins on trees
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when \tanhβ≤(d-1)^{-1/2}, including equality, for d ≥ 3 and β ≥ 0. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question.
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MANUSCRIPT 01
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 \tanhβ≤(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…
Full source abstract
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 \tanhβ≤(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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