| import Mathlib |
| import OAI.Probability.SKGap.Stability.RootDiagonalTransfer |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius |
| variable {n : ℕ} |
| local instance matrixMeasurable : MeasurableSpace (Matrix (Fin n) (Fin n) ℝ) := borel _ |
| local instance matrixBorel : BorelSpace (Matrix (Fin n) (Fin n) ℝ) := ⟨rfl⟩ |
|
|
| lemma continuous_joint_tapField (j : ℝ) : |
| Continuous (fun p : (Matrix (Fin n) (Fin n) ℝ × Field n) × Field n=>tapField j p.1.1 p.1.2 p.2) := by |
| unfold tapField onsager overlap magnetization |
| fun_prop |
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|
| lemma rootBad_iff_unplanted (j A ε c ρ : ℝ) (J : Matrix (Fin n) (Fin n) ℝ) (h : Field n) : |
| rootBad j A ε c ρ J h ↔ ∃ y : Field n,vectorNorm (tapField j J h y) ≤ ρ*sqrt (n:ℝ) ∧ |
| (J-(j/(n:ℝ)) • vecMulVec (fun _=>1) (fun _=>1),y)∈badFieldSet n j A ε c := by |
| simp only [rootBad,badFieldSet,mem_ofPred_eq,plantedInteraction,sub_add_cancel] |
|
|
| lemma isClosed_bounded_rootBad (j A ε c ρ : ℝ) (m : ℕ) : |
| IsClosed {p : Matrix (Fin n) (Fin n) ℝ × Field n | ∃ y ∈ Metric.closedBall (0 : Field n) m, |
| vectorNorm (tapField j p.1 p.2 y) ≤ ρ*sqrt (n:ℝ) ∧ |
| (p.1-(j/(n:ℝ)) • vecMulVec (fun _=>1) (fun _=>1),y)∈badFieldSet n j A ε c} := by |
| let X := ↥(Metric.closedBall (0 : Field n) m) |
| let : CompactSpace X := isCompact_iff_compactSpace.mp (isCompact_closedBall (0 : Field n) m) |
| let S : Set (X × (Matrix (Fin n) (Fin n) ℝ × Field n)) := |
| {p | vectorNorm (tapField j p.2.1 p.2.2 p.1) ≤ ρ*sqrt (n:ℝ) ∧ |
| (p.2.1-(j/(n:ℝ)) • vecMulVec (fun _=>1) (fun _=>1),(p.1:Field n))∈badFieldSet n j A ε c} |
| have hy : Continuous (fun p : X × (Matrix (Fin n) (Fin n) ℝ × Field n)=>(p.1:Field n)) := |
| continuous_subtype_val.comp continuous_fst |
| have ht := (continuous_joint_tapField (n:=n) j).comp (continuous_snd.prodMk hy) |
| have hS : IsClosed S := (isClosed_le ((continuous_vectorNorm n).comp ht) continuous_const).inter |
| ((isClosed_badFieldSet n j A ε c).preimage |
| (((continuous_fst.comp continuous_snd).sub continuous_const).prodMk hy)) |
| have hp := isClosedMap_snd_of_compactSpace S hS |
| convert hp using 1 |
| ext p |
| constructor |
| · rintro ⟨y,hy,hr,hb⟩ |
| exact ⟨(⟨y,hy⟩,p),⟨hr,hb⟩,rfl⟩ |
| · rintro ⟨⟨y,p⟩,⟨hr,hb⟩,rfl⟩ |
| exact ⟨y,y.property,hr,hb⟩ |
|
|
| theorem measurableSet_rootBad (j A ε c ρ : ℝ) : |
| MeasurableSet {p : Matrix (Fin n) (Fin n) ℝ × Field n | rootBad j A ε c ρ p.1 p.2} := by |
| have he : {p : Matrix (Fin n) (Fin n) ℝ × Field n | rootBad j A ε c ρ p.1 p.2}= |
| ⋃ m : ℕ,{p : Matrix (Fin n) (Fin n) ℝ × Field n | ∃ y ∈ Metric.closedBall (0 : Field n) m, |
| vectorNorm (tapField j p.1 p.2 y) ≤ ρ*sqrt (n:ℝ) ∧ |
| (p.1-(j/(n:ℝ)) • vecMulVec (fun _=>1) (fun _=>1),y)∈badFieldSet n j A ε c} := by |
| ext p |
| rw [mem_ofPred_eq,rootBad_iff_unplanted] |
| constructor |
| · rintro ⟨y,hr,hb⟩ |
| obtain ⟨m,hm⟩ := exists_nat_ge ‖y‖ |
| exact mem_iUnion.mpr ⟨m,y,by simpa only [Metric.mem_closedBall,dist_zero_right] using hm,hr,hb⟩ |
| · intro hp |
| obtain ⟨m,y,_,hr,hb⟩ := mem_iUnion.mp hp |
| exact ⟨y,hr,hb⟩ |
| rw [he] |
| exact MeasurableSet.iUnion (fun m=>(isClosed_bounded_rootBad j A ε c ρ m).measurableSet) |
| end SKGap |
| end |
| end |
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| end OAI |
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