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Exact square-lattice FK critical point and subcritical decay

Statement

Assume Countable Choice. For every real $q \geq 1$ , put $p_{s d} (q) = \sqrt{q} / (1 + \sqrt{q})$ . For every $p > p_{s d} (q)$ , the wired state has $θ^{1} (p, q) > 0$ ; for every $p < p_{s d} (q)$ , both free and wired states have zero percolation probability and the free and wired critical thresholds $p_{c}^{0} (q), p_{c}^{1} (q)$ equal $p_{s d} (q)$ . For every $p < p_{s d} (q)$ , there are constants $C (p, q), c (p, q) > 0$ such that every cylinder-limit FK state $ϕ$ on $Z^{2}$ satisfies $ϕ (x \leftrightarrow y) \leq C (p, q) e^{- c (p, q) ∥ x - y ∥_{2}} (x, y \in Z^{2}) .$ No equality of the free and wired measures at $p = p_{s d} (q)$ , critical exponent, or order of transition is asserted.

Statement references

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Free, wired, and subsequential infinite-volume FK states on the square lattice

Crossing sharpness and subcritical decay for square-lattice FK

Facts & Assumptions

Given: Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), real $q \geq 1$ , $p \in (0, 1)$ , the square-lattice free and wired cylinder-limit FK states, and the crossing/sharpness interface of Crossing sharpness and subcritical decay for square-lattice FK.

[F1]

For $b \in {0, 1}$ , $θ^{b} (p, q)$ and $p_{c}^{b} (q) = \inf {p : θ^{b} (p, q) > 0}$ have the definitions in Free, wired, and subsequential infinite-volume FK states on the square lattice.

[F2]

Wired positivity above $p_{s d}$ , zero free and wired percolation below $p_{s d}$ , subcritical exponential decay for every cylinder-limit state, and free/wired threshold coincidence are proved in Crossing sharpness and subcritical decay for square-lattice FK.

Proof

1.1F1F2algebra

Fix $q \geq 1$ and write $p_{s} = p_{s d} (q)$ . By [F2], $θ^{1} (p, q) > 0$ for every $p > p_{s}$ and $θ^{1} (p, q) = 0$ for every $p < p_{s}$ . Therefore the infimum defining $p_{c}^{1} (q)$ is $p_{s}$ : no parameter below $p_{s}$ belongs to the defining set, while every parameter above $p_{s}$ does.

2.1F1F2step 1.1

The free/wired threshold-coincidence conclusion in [F2] gives $p_{c}^{0} (q) = p_{c}^{1} (q) = p_{s}$ . This is a statement about the two infima; it does not identify the measures or their percolation probabilities at $p = p_{s}$ .

3.1F1F2∎

For $p < p_{s}$ , [F2] directly supplies constants $C (p, q), c (p, q) > 0$ for which the displayed two-point bound holds uniformly for every arbitrary-boundary cylinder-limit FK state. These are exactly the cylinder-limit states in the statement, with the definition in [F1]. The stated choice hypothesis is the one used by the supplier to construct the probability laws and its countable subcritical coupling.

Remarks

The threshold and subcritical conclusions apply to all real $q \geq 1$ . Countable Choice binds the actual infinite-volume probability-law and coupling construction; the underlying finite numerical threshold result has a separate choice-free local proof. No assertion at criticality is added.

Exact square-lattice FK critical point and subcritical decay

Statement

Statement references

Facts & Assumptions

Proof

Remarks

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