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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | import OAI.Probability.FreeIsing.Estimates.FairSpin
import OAI.Probability.FreeIsing.Trees.Cardinality
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
lemma freeCylinderWeight_zero_of_card {d : ℕ} (D : Finset (TreeVertex d))
(hcard : (inducedEdges D).card + 1 = D.card) (eta : TreeVertex d → IsingSpin) :
freeCylinderWeight 0 D eta = (1 / 2 : ENNReal) ^ D.card := by
simp only [freeCylinderWeight, neg_zero, ite_self, add_zero, Finset.prod_const]
rw [← pow_succ', hcard, ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 1 / 2)]
norm_num [ENNReal.ofReal_div_of_pos]
lemma fairSpinFactor_hasFreeIsingLaw_of_card {d : ℕ}
(hcard : ∀ D : Finset (TreeVertex d), ConnectedFinset D →
(inducedEdges D).card + 1 = D.card) :
HasFreeIsingLaw 0 (Measure.map (fairSpinFactor d) (labelMeasure d)) := by
intro D hD eta
rw [fairSpinFactor_cylinder, freeCylinderWeight_zero_of_card D (hcard D hD)]
lemma isFactorOfIID_zero_of_card (d : ℕ)
(hcard : ∀ D : Finset (TreeVertex d), ConnectedFinset D →
(inducedEdges D).card + 1 = D.card) : IsFactorOfIID d 0 := by
refine ⟨fairSpinFactor d, measurable_fairSpinFactor d,
fairSpinFactor_hasFreeIsingLaw_of_card hcard, ?_⟩
intro g
exact Filter.Eventually.of_forall (fairSpinFactor_relabel d g)
theorem isFactorOfIID_zero (d : ℕ) : IsFactorOfIID d 0 := by
exact isFactorOfIID_zero_of_card d (fun _ hD => hD.inducedEdges_card_add_one)
theorem free_ising_factor_beta_zero (d : ℕ) : IsFactorOfIID d (Real.tanh 0) := by
simpa only [Real.tanh_zero] using isFactorOfIID_zero d
end Problem367
end
end OAI
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