OAI/Probability/InvariantIsing/Cavity/CavityCylinderPenalty.lean

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3 declarations, 0 contain sorry, 0 declared as axioms

1. lemma OAI.InvariantIsing.finite_cylinder_penalty_comparison line 15
2. lemma OAI.InvariantIsing.cylinder_log_mean_tag line 67
3. theorem OAI.InvariantIsing.finite_cylinder_covariance_penalty_comparison line 84

   Source documentation

   Only the two covariance kernels matter: disjoint Gaussian copies automatically supply the zero mixed covariance required by interpolation.

   theorem finite_cylinder_covariance_penalty_comparison {X : Type*}
       [Fintype X] [MeasurableSpace X] [MeasurableSingletonClass X]
       (ν : Measure X) [IsProbabilityMeasure ν] (H V : X → ℝ) (c : ℝ)
       (C A : X → ℕ →₀ ℝ)
       (hK : ∀ x y, |cylinderCross (A x) (A y) - cylinderCross (C x) (C y)| ≤
         c * (V x + V y)) :
       (∫ g : ℕ → ℝ, Real.log (∫ x, Real.exp (H x - 2 * c * V x + cylinderField (C x) g) ∂ν)
         ∂gaussianCoordinates) ≤
       ∫ g : ℕ → ℝ, Real.log (∫ x, Real.exp (H x + cylinderField (A x) g) ∂ν)
         ∂gaussianCoordinates := by
   …