| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.BlockMassValue |
| import OAI.Probability.SKBarriers.Interpolation.SKHamiltonianDerivative |
| import OAI.Probability.SKBarriers.Gaussian.GaussianStepAlgebra |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| attribute [local instance 1900] cascadeNormedGroup cascadeNormedSpace |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
|
|
| section Generic |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| theorem cascadePressure_root (b : ℕ) (m : Fin b → ℝ) (g : E → ℝ) : |
| cascadePressure b m (fun z => g (cascadeRoot (E := E) b z)) = g := by |
| induction b with |
| | zero => rfl |
| | succ b ih => |
| change cascadePressure b _ (gaussianStep _ (fun z => g (cascadeRoot (E := E) b z.1))) = g |
| rw [gaussianStep_prefix (fun z => g (cascadeRoot (E := E) b z)) (m (Fin.last b)),ih] |
| end Generic |
|
|
| theorem coordinateProjection_append_prefix (a b : ℕ) (i : Fin a) |
| (z : CascadeSpace (ParameterSpace a) b) : |
| coordinateProjection (a+b) (Fin.castAdd b i) (parameterAppend a b z) = |
| coordinateProjection a i (cascadeRoot (E := ParameterSpace a) b z) := by |
| induction b with |
| | zero => rfl |
| | succ b ih => |
| change coordinateProjection ((a+b)+1) (Fin.castSucc (Fin.castAdd b i)) |
| (parameterAppend a b z.1,z.2) = _ |
| rw [coordinateProjection_castSucc,ih] |
| rfl |
|
|
| theorem blockExponent_zeroFields_append {D N k : ℕ} (I : Fin D → Finset (Fin N)) |
| (a : Fin D → ℝ) (s : Config N) (z : CascadeSpace (ParameterSpace D) ((k+1)*N)) : |
| blockExponent I a (fun _ : Fin (k+1) => 0) s (parameterAppend D ((k+1)*N) z) = |
| spinExponent D a I s (cascadeRoot (E := ParameterSpace D) ((k+1)*N) z) := by |
| rw [blockExponent_expansion] |
| simp only [zero_mul,Finset.sum_const_zero,add_zero,coordinateProjection_append_prefix, |
| spinExponent,coordinateLinear_apply] |
|
|
| theorem blockMass_at_disorder (D N k : ℕ) (i : Fin D) : |
| blockMass D N k (Fin.castAdd ((k+1)*N) i) = 0 := by |
| rw [blockMass_eq_sum] |
| apply Finset.sum_eq_zero |
| intro b _ |
| rw [ite_eq_right] |
| simp only [blockLevel_val,Fin.val_castAdd] |
| omega |
|
|
| theorem skBlockRoot_zero_fields (N k : ℕ) (β : ℝ) : |
| skBlockRoot N k β (fun _ => 0) = |
| ∫ z, affineLogPartition (fun _ : Config N => 0) |
| (spinExponent (Fintype.card (Edge N)) (fun _ => β/Real.sqrt (N:ℝ)) (skInteraction N)) z |
| ∂fiberGaussian (Fintype.card (Edge N)) 0 := by |
| unfold skBlockRoot |
| rw [hierarchyPressure_split] |
| have he : (fun z => affineLogPartition (fun _ : Config N => 0) |
| (blockExponent (skInteraction N) (fun _ => β/Real.sqrt (N:ℝ)) (fun _ : Fin (k+1) => 0)) |
| (parameterAppend (Fintype.card (Edge N)) ((k+1)*N) z)) = |
| fun z => affineLogPartition (fun _ : Config N => 0) |
| (spinExponent (Fintype.card (Edge N)) (fun _ => β/Real.sqrt (N:ℝ)) (skInteraction N)) |
| (cascadeRoot (E := ParameterSpace (Fintype.card (Edge N))) ((k+1)*N) z) := by |
| funext z |
| simp only [affineLogPartition,blockExponent_zeroFields_append] |
| rw [he,cascadePressure_root] |
| simp only [blockMass_at_disorder] |
| apply hierarchyPressure_zero |
| exact affineLogPartition_boundedDerivs _ _ |
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
