The exact factor-of-IID threshold for free Ising spins on trees

1 · The essentials

Imagine an endless branching network in which every point has the same number of neighbors and there are no loops. Each point carries a spin with one of two possible values, and neighboring spins favor agreement. The question is whether the particular pattern of randomness called the free Ising state can be generated from independent random labels placed at the points, without choosing a starting point or breaking the network’s symmetries. For three or more neighbors per point, the source gives an exact condition for when this is possible. If each point has $d$ neighbors and the interaction is controlled by $β \geq 0$ , the condition is $tanh β \leq (d - 1)^{- 1/2}$ . The boundary case, where equality holds, is included. The main result has a Lean formalization.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine identical households on an endless branching road network, each receiving an independently drawn random ticket. The question is whether one shared, symmetry-respecting rule can turn those tickets into two-color house decorations with exactly the prescribed tendency for neighboring houses to match. This image does not mean that each household needs to inspect only nearby tickets.

Why it matters

The result identifies exactly when independent randomness can generate the free ferromagnetic Ising state while respecting the tree’s symmetries. According to the source, it resolves the ferromagnetic case of Lyons’s question and the conjecture of Nam, Sly and Zhang. Including equality settles the boundary rather than leaving it undecided.

2 · Going further

The setting is the infinite $d$ -regular tree, with integer $d \geq 3$ , and the free zero-field ferromagnetic Ising measure at inverse temperature $β \geq 0$ . The question is whether this measure is a factor of IID: whether independent, identically distributed vertex labels can be transformed into spins with that measure by a rule respecting tree automorphisms. The exact criterion is $tanh β \leq (d - 1)^{- 1/2}$ .

The abstract identifies the positive implication, including equality, as the main contribution and says that the strict converse was already known. Thus the new direction supplies a spin factor throughout the stated range, while the known converse rules one out when $tanh β > (d - 1)^{- 1/2}$ . The repository summary describes this as resolving the ferromagnetic case of Lyons’s question; the abstract also identifies the positive implication as resolving the conjecture of Nam, Sly and Zhang.

The construction is described as requiring no root. The repository summary states almost-sure equivariance for each fixed automorphism, while the manuscript abstract states the stronger property that the spin factor can commute with every tree automorphism on every label input. The supplied abstract does not describe the construction’s internal steps, so a more detailed argument cannot be extracted from it. Only the main manuscript is supplied, marked Lean-formalized; no companion manuscripts or separate formalization-scope details are provided.

Useful notions

Trees and regular graphs
Independent, identically distributed randomness
Probability measures on spin configurations
Graph symmetries
The hyperbolic tangent function

What was known before / what is new

Before

The abstract says that the strict converse was known: above the stated threshold, the free Ising measure is not a factor of IID. It presents the positive implication, including the boundary, as the conjecture of Nam, Sly and Zhang. The repository summary places the result within the ferromagnetic case of Lyons’s question.

New

The main result supplies the positive implication for every integer $d \geq 3$ and every $β \geq 0$ satisfying the threshold, including equality. Together with the known strict converse, this gives an exact characterization. The manuscript abstract also states that the factor can respect every tree automorphism on every label input.

Limits and assumptions

The characterization concerns the free, zero-field, ferromagnetic Ising measure on infinite regular trees with $d \geq 3$ and $β \geq 0$ . It does not state conclusions for other graphs, other Ising states, or other parameter settings. The main result has a Lean formalization, but the supplied material does not specify whether every additional construction property is included in that formalization. Within the stated setting, the threshold question is settled, including equality; the source does not identify further open cases.

Mini glossary

Infinite d-regular tree: An infinite connected graph with no cycles in which every vertex has exactly $d$ neighbors.
Factor of IID: A random configuration obtained from independent, identically distributed labels by a rule respecting the graph’s symmetries.
Ferromagnetic Ising model: A model of two-valued spins in which interactions favor agreement between neighboring spins.
Zero field: The condition that no external field favors one spin value over the other.
Tree automorphism: A relabeling of a tree’s vertices that preserves which vertices are neighbors.
Equivariance: The property that applying a symmetry to the labels and then generating spins gives the same result as generating spins first and applying that symmetry afterward.

Keywords : Ising model · Factor of IID · Regular trees

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when $tanh β \leq (d - 1)^{- 1/2}$ , 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.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/236.md

Manuscripts in this family

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The sharp factor-of-IID threshold for the free Ising model on regular trees

26/09/2026

Verified by Lean

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 $tanh β \leq (d - 1)^{- 1/2}$ . 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.

PDF GitHub folder Lean formalization