| import Mathlib |
| import OAI.Probability.SKRatio.Calculus.PairUpdate |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators RealInnerProductSpace |
| namespace SKRatio.Fields |
| open Real Matrix TwoSpin |
| attribute [local instance] Classical.propDecidable |
| variable {n : ℕ} |
|
|
| lemma heatBathLinear_kernel (g : Disorder n) (h : Fin n→ℝ) (f : Spin n→ℝ) : |
| heatBathLinear g h f=0 ↔ ∀ x y,f x=f y := by |
| constructor |
| · intro hf |
| apply (dirichlet_eq_zero_iff g h f).mp |
| rw [← heatBathGenerator_form] |
| change expectation g h (fun x=>f x*(heatBathLinear g h f) x)=0 |
| rw [hf] |
| simp only [Pi.zero_apply,mul_zero,expectation_const] |
| · intro hf |
| have he : f=(fun _=>f (fun _=>false)) := funext (fun x=>hf x (fun _=>false)) |
| rw [he,heatBathLinear_const] |
|
|
| lemma heatBath_curvature_poincare (g : Disorder n) (h : Fin n→ℝ) |
| {ρ : ℝ} (hρ : 0 < ρ) |
| (hc : ∀ f : Spin n→ℝ,ρ*dirichlet g h f ≤ |
| expectation g h (fun x=>heatBathGenerator g h f x^2)) (f : Spin n→ℝ) : |
| ρ*variance g h f ≤ dirichlet g h f := by |
| let S := gibbsScale g h |
| let H : EuclideanSpace ℝ (Spin n) →ₗ[ℝ] EuclideanSpace ℝ (Spin n) := |
| S.toLinearMap ∘ₗ heatBathLinear g h ∘ₗ S.symm.toLinearMap |
| have hHS (k : Spin n→ℝ) : H (S k)=S (heatBathLinear g h k) := by simp [H] |
| have hH : H.IsSymmetric := by |
| intro u v |
| obtain ⟨u,rfl⟩ := S.surjective u |
| obtain ⟨v,rfl⟩ := S.surjective v |
| rw [hHS,hHS] |
| exact (gibbsScale_inner g h _ _).trans ((heatBathGenerator_adjoint g h u v).trans |
| (gibbsScale_inner g h _ _).symm) |
| have hform (k : Spin n→ℝ) : ⟪S k,H (S k)⟫=dirichlet g h k := by |
| rw [hHS,gibbsScale_inner] |
| exact heatBathGenerator_form g h k |
| have hpos : ∀ v,0 ≤ ⟪v,H v⟫ := by |
| intro v |
| obtain ⟨k,rfl⟩ := S.surjective v |
| rw [hform] |
| exact dirichlet_nonneg g h k |
| have hcurv : ∀ v,ρ*⟪v,H v⟫ ≤ ⟪H v,H v⟫ := by |
| intro v |
| obtain ⟨k,rfl⟩ := S.surjective v |
| rw [hform,hHS,gibbsScale_inner] |
| simpa only [sq,heatBathLinear,LinearMap.coe_mk,AddHom.coe_mk] using hc k |
| let f₀ := f-(fun _=>expectation g h f) |
| have hm : expectation g h f₀=0 := by |
| change expectation g h (fun x=>f x-expectation g h f)=0 |
| rw [expectation_sub,expectation_const,sub_self] |
| have hker : S f₀ ∈ (LinearMap.ker H)ᗮ := by |
| rw [Submodule.mem_orthogonal] |
| intro v hv |
| obtain ⟨k,rfl⟩ := S.surjective v |
| have hk : heatBathLinear g h k=0 := by |
| have hh := LinearMap.mem_ker.mp hv |
| rw [hHS] at hh |
| exact S.injective (by simpa using hh) |
| have hk' := (heatBathLinear_kernel g h k).mp hk |
| have he : k=(fun _=>k (fun _=>false)) := funext (fun x=>hk' x (fun _=>false)) |
| rw [gibbsScale_inner,he,expectation_const_mul,hm,mul_zero] |
| have hh := curvature_poincare_on_kernel_orthogonal H hH hpos hρ hcurv (S f₀) hker |
| rw [gibbsScale_inner,hform] at hh |
| have hd : dirichlet g h f₀=dirichlet g h f := by |
| rw [← heatBathGenerator_form,← heatBathGenerator_form] |
| have he : heatBathLinear g h f₀=heatBathLinear g h f := by |
| simp only [f₀,map_sub,heatBathLinear_const,sub_zero] |
| change expectation g h (fun x=>f₀ x*(heatBathLinear g h f₀) x)=_ |
| rw [he] |
| have hz : expectation g h (heatBathLinear g h f)=0 := by |
| change expectation g h (fun x=>∑ i,(f x-conditionalExpectation g h i f x))=0 |
| rw [expectation_sum] |
| apply Finset.sum_eq_zero |
| intro i _ |
| rw [expectation_sub,expectation_conditionalExpectation,sub_self] |
| simp only [f₀,Pi.sub_apply,sub_mul] |
| rw [expectation_sub,expectation_const_mul,hz,mul_zero,sub_zero] |
| rfl |
| rw [hd] at hh |
| simpa only [variance,f₀,Pi.sub_apply,sq] using hh |
|
|
| end SKRatio.Fields |
|
|
| end |
| end |
|
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| end OAI |
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|
