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Outline

Proof

Eigenfunction is Kakutani limit

We assume $ν$ is unique minimiser of $F(ν;φ) = ν(φ) - H(ν)$ among all probability measures and $μ$ minimises $F(μ;φ)$ among all translation invariant measures.

Then the Kakutani limit $$ h(x) = lim_n→\infty \frac{μ([x]_n)}{ν([x]_n)} $$ is an eigenfunction to $\mathcal{L}_φ$ provided it exists in $L^1(ν)$.

We can replace $μ$ with the twosided long-range FK-model

We can extend $μ$ and $ν$ to distributions on $(x,γ)∈ \X_± × \{0,1\}^\binom{\ZZ{2}}$ by taking $μ$ equal to the distribution long-range FK-model and $ν_- ⊗ ˜{η} ⊗ ν_+$ is the distribution of the random graph $(γ_-,ε,γ_+)$.

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