| import Mathlib |
| import OAI.Probability.SKBarriers.SpinGlass.SpinCoefficientVariation |
|
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
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| theorem hierarchyMeanOverlap_eq_mean_integrals {S : Type} [Fintype S] [Nonempty S] |
| {N : ℕ} (n : ℕ) (m : Fin n → ℝ) (U : S → ParameterSpace n →L[ℝ] ℝ) |
| (c : Fin N → S → ℝ) (hc : ∀ i s, ‖c i s‖ ≤ 1) (j : Fin (n+1)) : |
| hierarchyMeanOverlap n m U c j = |
| (∑ i, ∫ z, (hierarchySpinMean n m U (c i) j z)^2 |
| ∂hierarchyPathLaw n m (affineLogPartition (fun _ => 0) U) 0) / (N:ℝ) := by |
| unfold hierarchyMeanOverlap hierarchyMeanSquare normalizedSquare |
| rw [integral_div,integral_finsetSum] |
| intro i _ |
| have hr := hierarchySpinMean_regular n m U (c i) (by norm_num : (0:ℝ) ≤ 1) (hc i) j |
| apply hierarchyPathLaw_integrable n m _ _ (affineLogPartition_boundedDerivs (fun _ => 0) U) |
| (hr.1.pow 2) (C := 1) _ 0 |
| intro z |
| dsimp only [Pi.pow_apply] |
| rw [norm_pow] |
| exact pow_le_one₀ (norm_nonneg _) (hr.2 z) |
|
|
| theorem blockAtom_sum_mul (D N k : ℕ) (f : Fin (blockDimension D N k+1) → ℝ) : |
| (∑ j, hierarchyAtom (blockDimension D N k) (blockMass D N k) 1 j*f j) = |
| ∑ b : Fin (k+1), ((k+1 : ℕ) : ℝ)⁻¹*f (blockLevel D N k b) := by |
| simp only [hierarchyAtom_blockMass,blockAtom,Finset.sum_mul] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro b _ |
| simp only [ite_mul,zero_mul,Finset.sum_ite_eq,Finset.mem_univ,ite_true] |
|
|
| theorem fieldIndex_lt_blockLevel_iff (D N k : ℕ) (b l : Fin (k+1)) (i : Fin N) : |
| (fieldIndex D N k b i).val < (blockLevel D N k l).val ↔ b ≤ l := by |
| simp only [fieldIndex_val,blockLevel_val,Fin.le_def] |
| constructor |
| · intro h |
| by_contra hn |
| have hbl : l.val+1 ≤ b.val := by omega |
| have hmul := Nat.mul_le_mul_right N hbl |
| omega |
| · intro h |
| have hmul := Nat.mul_le_mul_right N h |
| nlinarith [i.isLt] |
|
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| theorem blockPressure_coordinate_variation {D N k : ℕ} (I : Fin D → Finset (Fin N)) |
| (a : Fin D → ℝ) (v : Fin (k+1) → ℝ) (b : Fin (k+1)) (i : Fin N) : |
| fderiv ℝ (fun c : Fin (blockDimension D N k) → ℝ => |
| hierarchyPressure (blockDimension D N k) (blockMass D N k) |
| (affineLogPartition (fun _ => 0) (spinExponent (blockDimension D N k) c (blockInteraction I))) 0) |
| (blockCoefficients a v) (Pi.single (fieldIndex D N k b i) 1) = |
| v b * (1-∑ l : Fin (k+1), if b ≤ l then ((k+1:ℕ):ℝ)⁻¹ * |
| (∫ z, (hierarchySpinMean (blockDimension D N k) (blockMass D N k) (blockExponent I a v) |
| (fun s => spin (s i)) (blockLevel D N k l) z)^2 |
| ∂hierarchyPathLaw (blockDimension D N k) (blockMass D N k) |
| (affineLogPartition (fun _ => 0) (blockExponent I a v)) 0) else 0) := by |
| rw [spinPressure_coefficient_variation,blockCoefficients_field,blockInteraction_field] |
| simp only [spinMonomial,Finset.prod_singleton] |
| congr 2 |
| have he (j : Fin (blockDimension D N k+1)) (c : ℝ) : |
| (if (fieldIndex D N k b i).val < j.val then |
| hierarchyAtom (blockDimension D N k) (blockMass D N k) 1 j*c else 0) = |
| hierarchyAtom (blockDimension D N k) (blockMass D N k) 1 j * |
| (if (fieldIndex D N k b i).val < j.val then c else 0) := by split_ifs <;> simp |
| simp_rw [he] |
| rw [blockAtom_sum_mul] |
| simp only [fieldIndex_lt_blockLevel_iff,mul_ite,mul_zero] |
| rfl |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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