openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Fields /FieldCanonicalVectorMoments.lean
| import OAI.Probability.InvariantIsing.Fields.FieldEndpointMoment | |
| namespace OAI | |
| /-! Independent canonical endpoint fields have a total second moment | |
| linear in the block size, uniformly over the cascade depth. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped BigOperators | |
| namespace InvariantIsing | |
| def fieldCanonicalVectorLaw (N : ℕ) (h : FieldStep) (b : Fin N → ℝ) : | |
| Measure (Fin N → ℝ) := | |
| Measure.pi (fun i => fieldCanonicalEndpointLaw h (b i)) | |
| instance fieldCanonicalVectorLaw_probability (N : ℕ) (h : FieldStep) (b : Fin N → ℝ) : | |
| IsProbabilityMeasure (fieldCanonicalVectorLaw N h b) := by | |
| unfold fieldCanonicalVectorLaw | |
| infer_instance | |
| lemma fieldCanonicalVectorLaw_coordinate_integrable {N : ℕ} (h : FieldStep) | |
| (b : Fin N → ℝ) (i : Fin N) : | |
| Integrable (fun z : Fin N → ℝ => z i ^ 2) (fieldCanonicalVectorLaw N h b) := | |
| (measurePreserving_eval (fun j => fieldCanonicalEndpointLaw h (b j)) i).integrable_comp_of_integrable | |
| (fieldCanonicalEndpointLaw_second_moment h (b i)).1 | |
| lemma fieldCanonicalVectorLaw_square_integrable {N : ℕ} (h : FieldStep) (b : Fin N → ℝ) : | |
| Integrable (fun z : Fin N → ℝ => ∑ i, z i ^ 2) (fieldCanonicalVectorLaw N h b) := by | |
| exact integrable_finsetSum _ (fun i _ => fieldCanonicalVectorLaw_coordinate_integrable h b i) | |
| theorem fieldCanonicalVectorLaw_second_moment {N : ℕ} (h : FieldStep) (b : Fin N → ℝ) | |
| {B H : ℝ} (hb : ∀ i, |b i| ≤ B) (hH : h.height (Fin.last h.depth) ≤ H) : | |
| (∫ z : Fin N → ℝ, ∑ i, z i ^ 2 ∂fieldCanonicalVectorLaw N h b) ≤ | |
| N * (2 * Real.exp (2 * B + 4 * H)) := by | |
| rw [integral_finsetSum _ (fun i _ => fieldCanonicalVectorLaw_coordinate_integrable h b i)] | |
| calc | |
| _ ≤ ∑ _i : Fin N, 2 * Real.exp (2 * B + 4 * H) := by | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| have he := (fieldCanonicalEndpointLaw_second_moment h (b i)).2 | |
| have hp : (∫ z : Fin N → ℝ, z i ^ 2 ∂fieldCanonicalVectorLaw N h b) = | |
| ∫ y : ℝ, y ^ 2 ∂fieldCanonicalEndpointLaw h (b i) := | |
| (measurePreserving_eval (fun j => fieldCanonicalEndpointLaw h (b j)) i).hasLaw.integral_comp | |
| (f := fun y : ℝ => y ^ 2) (by fun_prop) | |
| rw [hp] | |
| apply he.trans | |
| have hab := abs_le.mp (hb i) | |
| have h₁ : Real.exp (2 * b i + 4 * h.height (Fin.last h.depth)) ≤ | |
| Real.exp (2 * B + 4 * H) := Real.exp_le_exp.mpr (by linarith) | |
| have h₂ : Real.exp (-2 * b i + 4 * h.height (Fin.last h.depth)) ≤ | |
| Real.exp (2 * B + 4 * H) := Real.exp_le_exp.mpr (by linarith) | |
| linarith | |
| _ = _ := by simp | |
| end InvariantIsing | |
| end | |
| end OAI | |