| import Mathlib |
| import OAI.Probability.SKBarriers.Interpolation.ZeroFieldEndpoint |
| import OAI.Probability.SKBarriers.Hierarchy.WeightedBlockMass |
| import OAI.Probability.SKBarriers.Hierarchy.BlockObservable |
|
|
| namespace OAI |
|
|
| 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 |
| variable {S : Type} [Fintype S] [Nonempty S] |
|
|
| omit [Fintype S] [Nonempty S] in |
| theorem blockObservable_endpoint_one {D N k : ℕ} (H : Fin D → S → ℝ) |
| (F : Fin (k+1) → Fin N → S → ℝ) (s : S) (z : ParameterSpace (blockDimension D N k)) : |
| observableExponent (blockDimension D N k) (blockObservable H F) |
| (interpolationCoefficient (blockDimension D N k) (blockInterpolationChoice D N k) 1) s z= |
| ∑ i, H i s*coordinateProjection (blockDimension D N k) (Fin.castAdd ((k+1)*N) i) z := by |
| rw [observableExponent,coordinateLinear_apply,Fin.sum_univ_add] |
| simp only [interpolationCoefficient,blockInterpolationChoice,blockObservable,Fin.addCases_left, |
| Fin.addCases_right,ite_true,Bool.false_eq_true,ite_false,sub_self,Real.sqrt_zero,Real.sqrt_one, |
| one_mul,zero_mul,Finset.sum_const_zero,add_zero] |
|
|
| omit [Fintype S] [Nonempty S] in |
| theorem blockObservable_endpoint_zero {D N k : ℕ} (H : Fin D → S → ℝ) |
| (F : Fin (k+1) → Fin N → S → ℝ) (s : S) (z : ParameterSpace (blockDimension D N k)) : |
| observableExponent (blockDimension D N k) (blockObservable H F) |
| (interpolationCoefficient (blockDimension D N k) (blockInterpolationChoice D N k) 0) s z= |
| ∑ b, ∑ i, F b i s*coordinateProjection (blockDimension D N k) (fieldIndex D N k b i) z := by |
| rw [observableExponent,coordinateLinear_apply,Fin.sum_univ_add] |
| simp only [interpolationCoefficient,blockInterpolationChoice,blockObservable,Fin.addCases_left, |
| Fin.addCases_right,ite_true,Bool.false_eq_true,ite_false,sub_zero,Real.sqrt_zero,Real.sqrt_one, |
| one_mul,zero_mul,Finset.sum_const_zero,zero_add] |
| rw [← finProdFinEquiv.sum_comp (fun t : Fin ((k+1)*N) => |
| F (finProdFinEquiv.symm t).1 (finProdFinEquiv.symm t).2 s* |
| coordinateProjection (blockDimension D N k) (Fin.natAdd D t) z)] |
| simp only [Equiv.symm_apply_apply,Fintype.sum_prod_type,fieldIndex_eq_natAdd] |
|
|
| theorem blockObservable_root_one {D N k : ℕ} (H : Fin D → S → ℝ) |
| (F : Fin (k+1) → Fin N → S → ℝ) (c : S → ℝ) (w : Fin (k+1) → ℝ) : |
| hierarchyPressure (blockDimension D N k) (weightedBlockMass D N k w) |
| (affineLogPartition c (observableExponent (blockDimension D N k) (blockObservable H F) |
| (interpolationCoefficient (blockDimension D N k) (blockInterpolationChoice D N k) 1))) 0= |
| ∫ z, affineLogPartition c (fun s => coordinateLinear D (fun i => H i s)) z ∂fiberGaussian D 0 := by |
| rw [hierarchyPressure_split] |
| have he : (fun z => affineLogPartition c |
| (observableExponent (blockDimension D N k) (blockObservable H F) |
| (interpolationCoefficient (blockDimension D N k) (blockInterpolationChoice D N k) 1)) |
| (parameterAppend D ((k+1)*N) z))= |
| fun z => affineLogPartition c (fun s => coordinateLinear D (fun i => H i s)) |
| (cascadeRoot (E:=ParameterSpace D) ((k+1)*N) z) := by |
| funext z |
| simp only [affineLogPartition,blockObservable_endpoint_one,coordinateProjection_append_prefix, |
| coordinateLinear_apply] |
| rw [he,cascadePressure_root] |
| simp only [weightedBlockMass_at_disorder] |
| exact hierarchyPressure_zero D _ (affineLogPartition_boundedDerivs _ _) 0 |
| end |
| end SK.Analytic |
|
|
| end |
| end |
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| end OAI |
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