File size: 3,691 Bytes
135052a | 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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 | 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
end OAI
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