| import Mathlib |
| import OAI.Probability.SKGap.Localization.ScalarEigenfunction |
| |
| namespace OAI |
|
|
| section |
| open scoped BigOperators |
| namespace SKGapCutoff |
|
|
| noncomputable def centered {n : ℕ} (J : Interaction n) (f : Observables n) : Observables n := |
| fun x => f x - gibbsExpectation J f |
|
|
| @[simp] lemma gibbsExpectation_centered {n : ℕ} (J : Interaction n) (f : Observables n) : |
| gibbsExpectation J (centered J f) = 0 := by |
| unfold centered |
| rw [gibbsExpectation_sub, gibbsExpectation_const, sub_self] |
| |
| @[simp] lemma halfDiff_centered {n : ℕ} (J : Interaction n) (f : Observables n) |
| (i : Fin n) (x : Spin n) : halfDiff i (centered J f) x = halfDiff i f x := by |
| simp only [halfDiff, centered] |
| ring |
|
|
| lemma dirichlet_centered {n : ℕ} (J : Interaction n) (f g : Observables n) : |
| dirichlet J (centered J f) (centered J g) = dirichlet J f g := by |
| simp only [dirichlet, halfDiff_centered] |
|
|
| lemma centered_variance {n : ℕ} (J : Interaction n) (f : Observables n) : |
| stationaryInner J (centered J f) (centered J f) = gibbsVariance J f := by |
| simp only [stationaryInner, centered, gibbsVariance, pow_two] |
|
|
| lemma spectral_dirichlet {n : ℕ} (J : Interaction n) |
| (hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) (f : Observables n) : |
| dirichlet J f f = ∑ a, spinEigenvalues J hJ hdiag a * |
| ((spinEigenbasis J hJ hdiag).repr (gibbsEuclideanEquiv J f) a) ^ 2 := by |
| rw [← gibbs_dirichlet J hJ hdiag] |
| change stationaryInner J f (-generator J f) = _ |
| rw [← spectral_parseval J hJ hdiag] |
| simp_rw [spectral_generator_coordinates] |
| apply Finset.sum_congr rfl |
| intro a _ |
| ring |
|
|
| lemma centered_spectral_coordinate_zero {n : ℕ} (J : Interaction n) |
| (hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) (f : Observables n) |
| (a : Fin (Fintype.card (Spin n))) (ha : spinEigenvalues J hJ hdiag a = 0) : |
| (spinEigenbasis J hJ hdiag).repr (gibbsEuclideanEquiv J (centered J f)) a = 0 := by |
| have hg : generator J (scalarEigenfunction J hJ hdiag a) = 0 := by |
| rw [generator_scalarEigenfunction, ha, neg_zero, zero_smul] |
| have hc (x : Spin n) : scalarEigenfunction J hJ hdiag a x = |
| scalarEigenfunction J hJ hdiag a (fun _ => false) := |
| constant_of_generator_zero J hJ hdiag _ hg x _ |
| rw [spectral_coordinates_inner] |
| simp only [stationaryInner, hc, gibbsExpectation_mul_const, gibbsExpectation_centered, |
| mul_zero] |
| |
| lemma hasGap_of_positive_eigenvalues {n : ℕ} (J : Interaction n) |
| (hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) (γ : ℝ) |
| (hγ : ∀ a, 0 < spinEigenvalues J hJ hdiag a → γ ≤ spinEigenvalues J hJ hdiag a) : |
| HasGap J γ := by |
| intro f |
| rw [← centered_variance, ← dirichlet_centered J f f, spectral_dirichlet J hJ hdiag, |
| ← spectral_parseval J hJ hdiag, Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro a _ |
| by_cases ha : spinEigenvalues J hJ hdiag a = 0 |
| · simp only [centered_spectral_coordinate_zero J hJ hdiag f a ha, mul_zero, zero_pow, ne_eq, |
| OfNat.ofNat_ne_zero, not_false_eq_true, le_refl] |
| · have hp := lt_of_le_of_ne (spinEigenvalues_nonneg J hJ hdiag a) (Ne.symm ha) |
| simpa only [pow_two, mul_assoc] using |
| mul_le_mul_of_nonneg_right (hγ a hp) |
| (sq_nonneg ((spinEigenbasis J hJ hdiag).repr (gibbsEuclideanEquiv J (centered J f)) a)) |
|
|
| theorem finite_gap_pos {n : ℕ} (J : Interaction n) |
| (hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) : |
| ∃ γ : ℝ, 0 < γ ∧ HasGap J γ := by |
| let values : Finset ℝ := insert 1 ((Finset.univ.image (spinEigenvalues J hJ hdiag)).filter (0 < ·)) |
| have hne : values.Nonempty := ⟨1, Finset.mem_insert_self _ _⟩ |
| refine ⟨values.min' hne, ?_, hasGap_of_positive_eigenvalues J hJ hdiag _ ?_⟩ |
| · have hm := Finset.min'_mem values hne |
| rcases Finset.mem_insert.mp hm with h | h |
| · rw [h]; norm_num |
| · exact (Finset.mem_filter.mp h).2 |
| · intro a ha |
| apply Finset.min'_le |
| exact Finset.mem_insert_of_mem (Finset.mem_filter.mpr |
| ⟨Finset.mem_image.mpr ⟨a, Finset.mem_univ _, rfl⟩, ha⟩) |
| |
| end SKGapCutoff |
| |
| namespace SKGapCutoff |
| |
| lemma scalarEigenfunction_centered {n : ℕ} (J : Interaction n) |
| (hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) |
| (a : Fin (Fintype.card (Spin n))) (ha : spinEigenvalues J hJ hdiag a ≠ 0) : |
| gibbsExpectation J (scalarEigenfunction J hJ hdiag a) = 0 := by |
| have hz := gibbs_generator_zero J hJ hdiag (scalarEigenfunction J hJ hdiag a) |
| rw [generator_scalarEigenfunction] at hz |
| change gibbsExpectation J (fun x => -spinEigenvalues J hJ hdiag a * |
| scalarEigenfunction J hJ hdiag a x) = 0 at hz |
| rw [gibbsExpectation_mul_const] at hz |
| exact (mul_eq_zero.mp hz).resolve_left (neg_ne_zero.mpr ha) |
| |
| lemma scalarEigenfunction_inner_self {n : ℕ} (J : Interaction n) |
| (hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) |
| (a : Fin (Fintype.card (Spin n))) : |
| stationaryInner J (scalarEigenfunction J hJ hdiag a) |
| (scalarEigenfunction J hJ hdiag a) = 1 := by |
| rw [← gibbsEuclideanEquiv_inner, lift_scalarEigenfunction, |
| real_inner_self_eq_norm_sq, (spinEigenbasis J hJ hdiag).orthonormal.norm_eq_one, one_pow] |
| |
| lemma gap_le_positive_eigenvalue {n : ℕ} (J : Interaction n) |
| (hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) |
| {γ : ℝ} (hγ : HasGap J γ) (a : Fin (Fintype.card (Spin n))) |
| (ha : 0 < spinEigenvalues J hJ hdiag a) : γ ≤ spinEigenvalues J hJ hdiag a := by |
| have hv : gibbsVariance J (scalarEigenfunction J hJ hdiag a) = 1 := by |
| rw [gibbsVariance, scalarEigenfunction_centered J hJ hdiag a (ne_of_gt ha)] |
| simp only [sub_zero, pow_two] |
| exact scalarEigenfunction_inner_self J hJ hdiag a |
| have hd : dirichlet J (scalarEigenfunction J hJ hdiag a) |
| (scalarEigenfunction J hJ hdiag a) = spinEigenvalues J hJ hdiag a := by |
| rw [← gibbs_dirichlet J hJ hdiag, generator_scalarEigenfunction] |
| simp only [Pi.smul_apply, smul_eq_mul, neg_mul, neg_neg] |
| simp_rw [mul_left_comm (scalarEigenfunction J hJ hdiag a _) (spinEigenvalues J hJ hdiag a)] |
| rw [gibbsExpectation_mul_const] |
| change spinEigenvalues J hJ hdiag a * stationaryInner J |
| (scalarEigenfunction J hJ hdiag a) (scalarEigenfunction J hJ hdiag a) = _ |
| rw [scalarEigenfunction_inner_self, mul_one] |
| simpa only [hv, hd, mul_one] using hγ (scalarEigenfunction J hJ hdiag a) |
| |
| theorem hasGap_iff_positive_eigenvalues {n : ℕ} (J : Interaction n) |
| (hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) (γ : ℝ) : |
| HasGap J γ ↔ ∀ a, 0 < spinEigenvalues J hJ hdiag a → |
| γ ≤ spinEigenvalues J hJ hdiag a := |
| ⟨fun h a ha => gap_le_positive_eigenvalue J hJ hdiag h a ha, |
| hasGap_of_positive_eigenvalues J hJ hdiag γ⟩ |
| |
| theorem quenchedGap_iff_spectral (β : ℝ) : QuenchedGap β ↔ |
| ∃ γ : ℝ, 0 < γ ∧ Filter.Tendsto (fun n => (disorderLaw β n) |
| {g | ∀ a, 0 < spinEigenvalues (sampledInteraction g) |
| (sampledInteraction_symm g) (sampledInteraction_diag g) a → |
| γ ≤ spinEigenvalues (sampledInteraction g) |
| (sampledInteraction_symm g) (sampledInteraction_diag g) a}) |
| Filter.atTop (nhds 1) := by |
| simp only [QuenchedGap, hasGap_iff_positive_eigenvalues (sampledInteraction _) |
| (sampledInteraction_symm _) (sampledInteraction_diag _)] |
|
|
| end SKGapCutoff |
|
|
| end |
|
|
| end OAI |
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|
