| import Mathlib |
| import OAI.Probability.SKGap.Localization.ScalarParametersJoined |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| def plantedInteraction {n : ℕ} (j : ℝ) (W : Matrix (Fin n) (Fin n) ℝ) : Matrix (Fin n) (Fin n) ℝ := |
| W+(j/(n:ℝ)) • vecMulVec (fun _=>1) (fun _=>1) |
|
|
| def badFieldSet (n : ℕ) (j A ε c : ℝ) : Set (Matrix (Fin n) (Fin n) ℝ × (Fin n → ℝ)) := |
| {p | ∃ a : Fin n → ℝ, (∀ i,a i ∈ Icc 0 A) ∧ |
| (∑ i,(a i-spinVariance p.2 i)^2) ≤ ε^2*(n:ℝ) ∧ |
| ∃ x : EuclideanSpace ℝ (Fin n), ‖x‖=1 ∧ |
| quadraticForm (fieldHessian j (plantedInteraction j p.1) p.2 a) x.ofLp ≤ c} |
|
|
| lemma continuous_fieldHessian {n : ℕ} (j : ℝ) : |
| Continuous (fun p : (Matrix (Fin n) (Fin n) ℝ × (Fin n → ℝ)) × (Fin n → ℝ)=> |
| fieldHessian j p.1.1 p.1.2 p.2) := by |
| unfold fieldHessian hessianCore onsager overlap magnetization |
| fun_prop |
|
|
| lemma isClosed_badFieldSet (n : ℕ) (j A ε c : ℝ) : IsClosed (badFieldSet n j A ε c) := by |
| let X := ↥(Set.Icc (0 : Fin n → ℝ) (fun _=>A)) × ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1) |
| let : CompactSpace ↥(Set.Icc (0 : Fin n → ℝ) (fun _=>A)) := |
| isCompact_iff_compactSpace.mp isCompact_Icc |
| let : CompactSpace ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1) := |
| isCompact_iff_compactSpace.mp (isCompact_sphere 0 1) |
| let S : Set (X × (Matrix (Fin n) (Fin n) ℝ × (Fin n → ℝ))) := |
| {p | (∑ i,((p.1.1:Fin n → ℝ) i-spinVariance p.2.2 i)^2) ≤ ε^2*(n:ℝ) ∧ |
| quadraticForm (fieldHessian j (plantedInteraction j p.2.1) p.2.2 p.1.1) (p.1.2:EuclideanSpace ℝ (Fin n)).ofLp ≤ c} |
| have ha (i : Fin n) : Continuous (fun p : X × (Matrix (Fin n) (Fin n) ℝ × (Fin n → ℝ)) => |
| (p.1.1:Fin n → ℝ) i) := |
| (continuous_apply i).comp (continuous_subtype_val.comp (continuous_fst.comp continuous_fst)) |
| have hx (i : Fin n) : Continuous (fun p : X × (Matrix (Fin n) (Fin n) ℝ × (Fin n → ℝ)) => |
| (p.1.2:EuclideanSpace ℝ (Fin n)).ofLp i) := |
| (continuous_apply i).comp ((PiLp.continuous_ofLp 2 (fun _ : Fin n=>ℝ)).comp |
| (continuous_subtype_val.comp (continuous_snd.comp continuous_fst))) |
| have hy (i : Fin n) : Continuous (fun p : X × (Matrix (Fin n) (Fin n) ℝ × (Fin n → ℝ)) => |
| tanh (p.2.2 i)) := continuous_tanh.comp ((continuous_apply i).comp (continuous_snd.comp continuous_snd)) |
| have hS : IsClosed S := by |
| apply IsClosed.inter |
| · exact isClosed_le (f := fun p : X × (Matrix (Fin n) (Fin n) ℝ × (Fin n → ℝ)) => |
| ∑ i,((p.1.1:Fin n → ℝ) i-spinVariance p.2.2 i)^2) |
| (g := fun _ => ε^2*(n:ℝ)) (by |
| apply continuous_finsetSum |
| intro i _ |
| exact ((ha i).sub (continuous_const.sub ((hy i).pow 2))).pow 2) continuous_const |
| · exact isClosed_le (f := fun p : X × (Matrix (Fin n) (Fin n) ℝ × (Fin n → ℝ)) => |
| quadraticForm (fieldHessian j (plantedInteraction j p.2.1) p.2.2 p.1.1) |
| (p.1.2:EuclideanSpace ℝ (Fin n)).ofLp) (g := fun _=>c) (by |
| dsimp [X,quadraticForm,fieldHessian,hessianCore,plantedInteraction,onsager,overlap,magnetization] |
| apply continuous_finsetSum; intro i _ |
| apply continuous_finsetSum; intro k _ |
| apply Continuous.mul _ (hx k) |
| apply Continuous.mul (hx i) |
| apply Continuous.add continuous_const |
| apply Continuous.mul _ (Real.continuous_sqrt.comp (ha k)) |
| apply Continuous.mul (Real.continuous_sqrt.comp (ha i)) |
| exact (((continuous_const.mul (continuous_const.sub |
| ((continuous_finsetSum _ (fun l _ => (hy l).pow 2)).div_const _))).mul continuous_const).sub |
| ((by fun_prop : Continuous (fun p : X × (Matrix (Fin n) (Fin n) ℝ × (Fin n → ℝ)) => p.2.1 i k)).add continuous_const)).sub |
| (((continuous_const.mul (hy i))).mul (hy k))) continuous_const |
| have hproj := isClosedMap_snd_of_compactSpace S hS |
| convert hproj using 1 |
| ext p |
| constructor |
| · rintro ⟨a,ha,hclose,x,hx,hquad⟩ |
| have ha' : a ∈ Icc (0 : Fin n → ℝ) (fun _=>A) := ⟨fun i=>(ha i).1,fun i=>(ha i).2⟩ |
| have hx' : x ∈ Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1 := by simpa only [Metric.mem_sphere,dist_zero_right] using hx |
| exact ⟨((⟨a,ha'⟩,⟨x,hx'⟩),p),⟨hclose,hquad⟩,rfl⟩ |
| · rintro ⟨⟨⟨a,x⟩,p'⟩,hS,rfl⟩ |
| exact ⟨a,fun i=>⟨a.2.1 i,a.2.2 i⟩,hS.1,x,by simpa only [Metric.mem_sphere,dist_zero_right] using x.2,hS.2⟩ |
| |
| end SKGap |
| end |
| end |
| |
| end OAI |
| |
