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Unique interacting Ising state at sufficient high temperature

Statement

Assume Countable Choice, spins $s_{i} \in {- 1, + 1}$ , inverse temperature $β > 0$ , a constant finite field $h \in R$ , and symmetric pair couplings with zero diagonal, $J_{i j} = J_{j i}$ and $J_{i i} = 0$ , satisfying $J_{i + k, j + k} = J_{i j}$ for all $i, j, k \in Z^{d}$ , and $J_{i j} = 0$ when $∥ i - j ∥_{\infty} > R$ for some finite integer $R \geq 0$ . For finite $Λ$ , use the conditional energy $E_{Λ} (s_{Λ} ∣ η) = - \sum_{{i, j} : {i, j} \cap Λ \neq \emptyset} J_{i j} {\hat{s}}_{i} {\hat{s}}_{j} - h \sum_{i \in Λ} s_{i},$ where ${\hat{s}}_{i} = s_{i}$ on $Λ$ and ${\hat{s}}_{i} = η_{i}$ outside; let $γ$ be the corresponding finite-alphabet Gibbs specification from Lattice Gibbs kernels and DLR states. If $\sup_{i} \sum_{j} \tanh (β ∣ J_{i j} ∣) < 1,$ then there exists exactly one DLR state. For homogeneous nearest-neighbor coupling $J$ this sufficient condition is $2 d \tanh (β ∣ J ∣) < 1$ . It is not asserted to be sharp.

Facts & Assumptions

Given: The finite-range Ising pair Hamiltonian and one-site conditional Gibbs laws in the Statement, with $β > 0$ and constant finite field $h$ .

[A1]

Assume Countable Choice: every countable family of nonempty sets admits a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Under that assumption, every proper consistent cylinder-local specification on this finite-alphabet product has a DLR state (Finite-range lattice DLR existence and closed convex state set).

[F1]

The one-site influence satisfies $c_{i j} \leq \tanh (β ∣ J_{i j} ∣)$ for this Hamiltonian normalization (Ising influence bounded by hyperbolic tangent).

[F2]

A finite-range specification with strict influence row sum $c < 1$ has exactly one DLR state (Finite-range Dobrushin uniqueness and boundary influence).

[F3]

The finite-range lattice Gibbs kernels are proper, consistent and finite range under the stated hypotheses (Conditional Gibbs kernel consistency).

[F4]

To realize the displayed pair Hamiltonian in the interaction-list convention of Finite-range lattice interaction and conditional energy, choose one displacement $v$ from each pair ${v, - v}$ with $J_{0 v} \neq 0$ , use shape $S_{v} = {0, v}$ and term $Φ_{v} (s_{0}, s_{v}) = - J_{0 v} s_{0} s_{v}$ , and add the onsite shape $S_{0} = {0}$ with $Φ_{0} (s_{0}) = - h s_{0}$ . Every unordered pair is counted once through its unique representative and translated anchor; this finite list has uniform one-site dependence radius at most $R$ .

Proof

1.1givenF1algebra

By [F1], $\sup_{i} \sum_{j} c_{i j} \leq \sup_{i} \sum_{j} \tanh (β ∣ J_{i j} ∣) < 1$ .

2.1A1F2F3F4step 1.1∎

The interaction list in [F4] has exactly the conditional energy displayed in the Statement: each unordered pair occurs once, and each site contributes the single field term $- h s_{i}$ . Thus [F3] gives its proper consistent finite-range Gibbs specification. Existence follows from [A1] for this specification. Apply [F2] with the uniform radius in [F4] and row-sum bound from step 1.1 to obtain uniqueness. This criterion is sufficient only.

Unique interacting Ising state at sufficient high temperature

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources