import Mathlib namespace OAI /-! The nearest-neighbor Anderson operator on the square lattice and its almost-sure spectral interval. -/ noncomputable section open MeasureTheory open scoped ENNReal namespace PlanarAnderson abbrev Site := ℤ × ℤ abbrev Configuration := Site → ℝ abbrev Hilbert := lp (fun _ : Site => ℂ) 2 instance : Nontrivial Hilbert := by refine ⟨⟨lp.single 2 (0, 0) (1 : ℂ), 0, ?_⟩⟩ intro he have h := congrArg (fun u : Hilbert => u (0, 0)) he simp at h def disorderLaw (h : ℝ) : Measure Configuration := Measure.infinitePi (fun _ : Site => ProbabilityTheory.cond volume (Set.Icc (-h) h)) def IsAndersonOperator (v : Configuration) (H : Hilbert →L[ℂ] Hilbert) : Prop := ∀ (u : Hilbert) (x : Site), (H u) x = u (x.1 + 1, x.2) + u (x.1 - 1, x.2) + u (x.1, x.2 + 1) + u (x.1, x.2 - 1) + (v x : ℂ) * u x def spectralInterval (h : ℝ) : Set ℂ := {z | z.im = 0 ∧ -4 - h ≤ z.re ∧ z.re ≤ 4 + h} def eigenvalues (H : Hilbert →L[ℂ] Hilbert) : Set ℂ := {z | ∃ u : Hilbert, u ≠ 0 ∧ H u = z • u} def PurePoint (H : Hilbert →L[ℂ] Hilbert) : Prop := (Submodule.span ℂ {u : Hilbert | ∃ z : ℂ, u ≠ 0 ∧ H u = z • u}).topologicalClosure = ⊤ def HasCompleteOrthonormalEigenbasis (H : Hilbert →L[ℂ] Hilbert) : Prop := ∃ b : HilbertBasis ℕ ℂ Hilbert, ∀ k, ∃ z : ℝ, H (b k) = (z : ℂ) • b k def MainStatement (h : ℝ) : Prop := ∀ᵐ v ∂disorderLaw h, ∃ H : Hilbert →L[ℂ] Hilbert, IsAndersonOperator v H ∧ PurePoint H ∧ HasCompleteOrthonormalEigenbasis H ∧ spectrum ℂ H = spectralInterval h ∧ closure (eigenvalues H) = spectralInterval h private theorem mem_reindex (e : Site ≃ Site) (u : Hilbert) : Memℓp (fun x => u (e x)) 2 := by apply memℓp_gen exact e.summable_iff.mpr ((lp.memℓp u).summable (by norm_num)) def permute (e : Site ≃ Site) : Hilbert →ₗᵢ[ℂ] Hilbert where toFun u := ⟨fun x => u (e x), mem_reindex e u⟩ map_add' u v := by ext; rfl map_smul' c u := by ext; rfl norm_map' u := by rw [lp.norm_eq_tsum_rpow (by norm_num), lp.norm_eq_tsum_rpow (by norm_num)] congr 1 exact e.tsum_eq (fun x => ‖u x‖ ^ (2 : ℝ≥0∞).toReal) @[simp] theorem permute_apply (e : Site ≃ Site) (u : Hilbert) (x : Site) : permute e u x = u (e x) := rfl def translation (a : Site) : Site ≃ Site where toFun x := x + a invFun x := x - a left_inv x := add_sub_cancel_right x a right_inv x := sub_add_cancel x a @[simp] theorem translation_apply (a x : Site) : translation a x = x + a := rfl theorem inner_permute (e : Site ≃ Site) (u w : Hilbert) : inner ℂ (permute e u) w = inner ℂ u (permute e.symm w) := by rw [lp.inner_eq_tsum, lp.inner_eq_tsum] conv_rhs => rw [← e.tsum_eq (fun x => inner ℂ (u x) (permute e.symm w x))] simp def hopping : Hilbert →L[ℂ] Hilbert := (permute (translation (1, 0))).toContinuousLinearMap + (permute (translation (-1, 0))).toContinuousLinearMap + (permute (translation (0, 1))).toContinuousLinearMap + (permute (translation (0, -1))).toContinuousLinearMap @[simp] theorem hopping_apply (u : Hilbert) (x : Site) : hopping u x = u (x.1 + 1, x.2) + u (x.1 - 1, x.2) + u (x.1, x.2 + 1) + u (x.1, x.2 - 1) := by simp [hopping, translation, sub_eq_add_neg, Prod.add_def] theorem norm_hopping_le : ‖hopping‖ ≤ 4 := by have he (a : Site) : ‖(permute (translation a)).toContinuousLinearMap‖ ≤ 1 := LinearIsometry.norm_toContinuousLinearMap_le _ have h1 := norm_add_le (permute (translation (1, 0))).toContinuousLinearMap (permute (translation (-1, 0))).toContinuousLinearMap have h2 := norm_add_le ((permute (translation (1, 0))).toContinuousLinearMap + (permute (translation (-1, 0))).toContinuousLinearMap) (permute (translation (0, 1))).toContinuousLinearMap have h3 := norm_add_le ((permute (translation (1, 0))).toContinuousLinearMap + (permute (translation (-1, 0))).toContinuousLinearMap + (permute (translation (0, 1))).toContinuousLinearMap) (permute (translation (0, -1))).toContinuousLinearMap dsimp [hopping] linarith [he (1, 0), he (-1, 0), he (0, 1), he (0, -1)] theorem translation_symm (a : Site) : (translation a).symm = translation (-a) := by apply Equiv.ext intro x change x - a = x + -a exact sub_eq_add_neg x a theorem hopping_selfAdjoint : IsSelfAdjoint hopping := by apply ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric.mpr intro u w change inner ℂ (hopping u) w = inner ℂ u (hopping w) simp only [hopping, add_apply, inner_add_left, inner_add_right, LinearIsometry.coe_toContinuousLinearMap, inner_permute, translation_symm] simp only [Prod.neg_mk, neg_neg, neg_zero] ring def potential (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) : Hilbert →L[ℂ] Hilbert := lp.mapCLM 2 (fun x => (v x : ℂ) • ContinuousLinearMap.id ℂ ℂ) hh (fun x => by simpa only [norm_smul, ContinuousLinearMap.norm_id, mul_one, Complex.norm_real, Real.norm_eq_abs] using hv x) @[simp] theorem potential_apply (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) (u : Hilbert) (x : Site) : potential v hh hv u x = (v x : ℂ) * u x := rfl theorem norm_potential_le (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) : ‖potential v hh hv‖ ≤ h := lp.norm_mapCLM_le _ _ _ _ theorem potential_selfAdjoint (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) : IsSelfAdjoint (potential v hh hv) := by apply ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric.mpr intro u w change inner ℂ (potential v hh hv u) w = inner ℂ u (potential v hh hv w) rw [lp.inner_eq_tsum, lp.inner_eq_tsum] congr 1 funext x simp [RCLike.inner_apply, mul_left_comm, mul_assoc] def anderson (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) : Hilbert →L[ℂ] Hilbert := hopping + potential v hh hv theorem anderson_formula (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) : IsAndersonOperator v (anderson v hh hv) := by intro u x simp [anderson] theorem anderson_selfAdjoint (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) : IsSelfAdjoint (anderson v hh hv) := by change IsSelfAdjoint (hopping + potential v hh hv) exact IsSelfAdjoint.add (R := Hilbert →L[ℂ] Hilbert) hopping_selfAdjoint (potential_selfAdjoint v hh hv) theorem norm_anderson_le (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) : ‖anderson v hh hv‖ ≤ 4 + h := (norm_add_le _ _).trans (add_le_add norm_hopping_le (norm_potential_le v hh hv)) theorem anderson_unique {v : Configuration} {H K : Hilbert →L[ℂ] Hilbert} (hH : IsAndersonOperator v H) (hK : IsAndersonOperator v K) : H = K := by ext u x rw [hH, hK] theorem spectrum_subset (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) : spectrum ℂ (anderson v hh hv) ⊆ spectralInterval h := by intro z hz have hr := (anderson_selfAdjoint v hh hv).im_eq_zero_of_mem_spectrum hz have hn := (spectrum.norm_le_norm_of_mem hz).trans (norm_anderson_le v hh hv) refine ⟨hr, ?_, ?_⟩ · have : -z.re ≤ ‖z‖ := by simpa using Complex.re_le_norm (-z) linarith · exact (Complex.re_le_norm z).trans hn theorem uniform_probability {h : ℝ} (hh : 0 < h) : IsProbabilityMeasure (ProbabilityTheory.cond volume (Set.Icc (-h) h)) := by apply ProbabilityTheory.cond_isProbabilityMeasure_of_finite · rw [Real.volume_Icc] exact ne_of_gt (ENNReal.ofReal_pos.mpr (by linarith)) · rw [Real.volume_Icc] exact ENNReal.ofReal_ne_top theorem bounded_ae {h : ℝ} (hh : 0 < h) : ∀ᵐ v ∂disorderLaw h, ∀ x, |v x| ≤ h := by let := uniform_probability hh rw [ae_all_iff] intro x have he := (measurePreserving_eval_infinitePi (fun _ : Site => ProbabilityTheory.cond volume (Set.Icc (-h) h)) x).quasiMeasurePreserving.ae (ProbabilityTheory.ae_cond_mem measurableSet_Icc) filter_upwards [he] with v hv exact abs_le.mpr hv theorem anderson_exists_ae {h : ℝ} (hh : 0 < h) : ∀ᵐ v ∂disorderLaw h, ∃ H : Hilbert →L[ℂ] Hilbert, IsAndersonOperator v H ∧ IsSelfAdjoint H ∧ spectrum ℂ H ⊆ spectralInterval h := by filter_upwards [bounded_ae hh] with v hv exact ⟨anderson v hh.le hv, anderson_formula v hh.le hv, anderson_selfAdjoint v hh.le hv, spectrum_subset v hh.le hv⟩ def cube (n : ℕ) : Finset Site := Finset.Icc 0 (n : ℤ) ×ˢ Finset.Icc 0 (n : ℤ) def outerCube (n : ℕ) : Finset Site := Finset.Icc (-1) ((n : ℤ) + 1) ×ˢ Finset.Icc (-1) ((n : ℤ) + 1) def innerCube (n : ℕ) : Finset Site := Finset.Icc 1 ((n : ℤ) - 1) ×ˢ Finset.Icc 1 ((n : ℤ) - 1) def boundary (n : ℕ) : Finset Site := outerCube n \ innerCube n def plane (q : ℂ) (x : Site) : ℂ := q ^ (x.1 + x.2) def cutWave (q : ℂ) (n : ℕ) : Hilbert := ∑ x ∈ cube n, lp.single 2 x (plane q x) def freeEnergy (q : ℂ) : ℂ := 2 * (q + q⁻¹) @[simp] theorem cutWave_apply (q : ℂ) (n : ℕ) (x : Site) : cutWave q n x = if x ∈ cube n then plane q x else 0 := by simp only [cutWave, lp.coeFn_sum, Finset.sum_apply, lp.coeFn_single, Finset.sum_pi_single] theorem plane_norm {q : ℂ} (hq : ‖q‖ = 1) (x : Site) : ‖plane q x‖ = 1 := by simp [plane, norm_zpow, hq] theorem cube_card (n : ℕ) : (cube n).card = (n + 1) ^ 2 := by simp [cube, Finset.card_product, Int.card_Icc, pow_two] theorem cutWave_norm {q : ℂ} (hq : ‖q‖ = 1) (n : ℕ) : ‖cutWave q n‖ = n + 1 := by have hsq : ‖cutWave q n‖ ^ 2 = ((n : ℝ) + 1) ^ 2 := by have hs := lp.norm_sum_single (E := fun _ : Site => ℂ) (p := 2) (by norm_num) (plane q) (cube n) norm_num only [ENNReal.toReal_ofNat, Real.rpow_two] at hs rw [show (∑ x ∈ cube n, ‖plane q x‖ ^ (2 : ℕ)) = (cube n).card by simp [plane_norm hq]] at hs simpa [cutWave, cube_card] using hs nlinarith [norm_nonneg (cutWave q n), (show (0 : ℝ) ≤ n by positivity)] theorem cutWave_point_bound {q : ℂ} (hq : ‖q‖ = 1) (n : ℕ) (x : Site) : ‖cutWave q n x‖ ≤ 1 := by rw [cutWave_apply] split <;> simp [plane_norm hq] theorem freeEnergy_norm_le {q : ℂ} (hq : ‖q‖ = 1) : ‖freeEnergy q‖ ≤ 4 := by dsimp [freeEnergy] rw [norm_mul] norm_num have hi : ‖q + q⁻¹‖ ≤ 2 := by simpa only [hq, norm_inv, inv_one, one_add_one_eq_two] using norm_add_le q q⁻¹ linarith theorem plane_hopping {q : ℂ} (hq : q ≠ 0) (x : Site) : plane q (x.1 + 1, x.2) + plane q (x.1 - 1, x.2) + plane q (x.1, x.2 + 1) + plane q (x.1, x.2 - 1) = freeEnergy q * plane q x := by have e1 : x.1 + 1 + x.2 = (x.1 + x.2) + 1 := by omega have e2 : x.1 - 1 + x.2 = (x.1 + x.2) + (-1) := by omega have e3 : x.1 + (x.2 + 1) = (x.1 + x.2) + 1 := by omega have e4 : x.1 + (x.2 - 1) = (x.1 + x.2) + (-1) := by omega simp only [plane, freeEnergy, e1, e2, e3, e4, zpow_add₀ hq, zpow_one, zpow_neg_one] ring theorem free_error_inner {q : ℂ} (hq : q ≠ 0) (n : ℕ) (x : Site) (hx : x ∈ innerCube n) : (hopping (cutWave q n) - freeEnergy q • cutWave q n) x = 0 := by simp only [innerCube, Finset.mem_product, Finset.mem_Icc] at hx have h0 : x ∈ cube n := by simp [cube, Finset.mem_product, Finset.mem_Icc]; omega have h1 : (x.1 + 1, x.2) ∈ cube n := by simp [cube, Finset.mem_product, Finset.mem_Icc]; omega have h2 : (x.1 - 1, x.2) ∈ cube n := by simp [cube, Finset.mem_product, Finset.mem_Icc]; omega have h3 : (x.1, x.2 + 1) ∈ cube n := by simp [cube, Finset.mem_product, Finset.mem_Icc]; omega have h4 : (x.1, x.2 - 1) ∈ cube n := by simp [cube, Finset.mem_product, Finset.mem_Icc]; omega simp only [lp.coeFn_sub, Pi.sub_apply, lp.coeFn_smul, Pi.smul_apply, smul_eq_mul, hopping_apply, cutWave_apply, ite_eq_left h0, ite_eq_left h1, ite_eq_left h2, ite_eq_left h3, ite_eq_left h4] exact sub_eq_zero.mpr (plane_hopping hq x) theorem free_error_outer (q : ℂ) (n : ℕ) (x : Site) (hx : x ∉ outerCube n) : (hopping (cutWave q n) - freeEnergy q • cutWave q n) x = 0 := by simp only [outerCube, Finset.mem_product, Finset.mem_Icc, not_and] at hx have h0 : x ∉ cube n := by simp [cube, Finset.mem_product, Finset.mem_Icc]; omega have h1 : (x.1 + 1, x.2) ∉ cube n := by simp [cube, Finset.mem_product, Finset.mem_Icc]; omega have h2 : (x.1 - 1, x.2) ∉ cube n := by simp [cube, Finset.mem_product, Finset.mem_Icc]; omega have h3 : (x.1, x.2 + 1) ∉ cube n := by simp [cube, Finset.mem_product, Finset.mem_Icc]; omega have h4 : (x.1, x.2 - 1) ∉ cube n := by simp [cube, Finset.mem_product, Finset.mem_Icc]; omega simp only [lp.coeFn_sub, Pi.sub_apply, lp.coeFn_smul, Pi.smul_apply, smul_eq_mul, hopping_apply, cutWave_apply, ite_eq_right h0, ite_eq_right h1, ite_eq_right h2, ite_eq_right h3, ite_eq_right h4] simp theorem boundary_card (n : ℕ) (hn : 1 ≤ n) : (boundary n).card = 8 * (n + 1) := by have hi : innerCube n ⊆ outerCube n := by intro x hx simp only [innerCube, outerCube, Finset.mem_product, Finset.mem_Icc] at hx ⊢ omega rw [boundary, Finset.card_sdiff_of_subset hi] have ho : (outerCube n).card = (n + 3)^2 := by simp only [outerCube, Finset.card_product, Int.card_Icc] have : (↑n + 1 + 1 - (-1) : ℤ).toNat = n + 3 := by omega rw [this, pow_two] have hj : (innerCube n).card = (n - 1)^2 := by simp only [innerCube, Finset.card_product, Int.card_Icc] have : (↑n - 1 + 1 - 1 : ℤ).toNat = n - 1 := by omega rw [this, pow_two] rw [ho, hj] have : (n + 3)^2 = (n - 1)^2 + 8 * (n + 1) := by obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hn simp only [Nat.add_sub_cancel_left] ring omega theorem free_error_point_bound {q : ℂ} (hq : ‖q‖ = 1) (n : ℕ) (x : Site) : ‖(hopping (cutWave q n) - freeEnergy q • cutWave q n) x‖ ≤ 8 := by simp only [lp.coeFn_sub, Pi.sub_apply, lp.coeFn_smul, Pi.smul_apply, smul_eq_mul, hopping_apply] calc _ ≤ ‖cutWave q n (x.1+1,x.2)‖ + ‖cutWave q n (x.1-1,x.2)‖ + ‖cutWave q n (x.1,x.2+1)‖ + ‖cutWave q n (x.1,x.2-1)‖ + ‖freeEnergy q‖ * ‖cutWave q n x‖ := by grw [norm_sub_le, norm_add_le, norm_add_le, norm_add_le, norm_mul] _ ≤ 8 := by have hm := mul_le_mul (freeEnergy_norm_le hq) (cutWave_point_bound hq n x) (norm_nonneg _) (by norm_num : (0:ℝ) ≤ 4) linarith [cutWave_point_bound hq n (x.1+1,x.2), cutWave_point_bound hq n (x.1-1,x.2), cutWave_point_bound hq n (x.1,x.2+1), cutWave_point_bound hq n (x.1,x.2-1)] theorem finite_norm_sq (u : Hilbert) (s : Finset Site) (hu : ∀ x ∉ s, u x = 0) : ‖u‖^2 = ∑ x ∈ s, ‖u x‖^2 := by have he := lp.norm_rpow_eq_tsum (p := 2) (by norm_num) u norm_num only [ENNReal.toReal_ofNat, Real.rpow_two] at he rw [he] apply tsum_eq_sum intro x hx simp [hu x hx] theorem free_error_norm_sq {q : ℂ} (hq : ‖q‖ = 1) (n : ℕ) (hn : 1 ≤ n) : ‖hopping (cutWave q n) - freeEnergy q • cutWave q n‖^2 ≤ 512 * ((n:ℝ)+1) := by have hq0 : q ≠ 0 := by intro he; simp [he] at hq have hs : ∀ x ∉ boundary n, (hopping (cutWave q n) - freeEnergy q • cutWave q n) x = 0 := by intro x hx by_cases ho : x ∈ outerCube n · have hi : x ∈ innerCube n := by simpa [boundary, ho] using hx exact free_error_inner hq0 n x hi · exact free_error_outer q n x ho rw [finite_norm_sq _ (boundary n) hs] calc _ ≤ ∑ _x ∈ boundary n, (64:ℝ) := by apply Finset.sum_le_sum intro x hx have hb := free_error_point_bound hq n x nlinarith [norm_nonneg ((hopping (cutWave q n) - freeEnergy q • cutWave q n) x)] _ = _ := by simp [boundary_card n hn]; ring theorem free_error_relative {q : ℂ} (hq : ‖q‖ = 1) {δ : ℝ} (hδ : 0 < δ) : ∃ n : ℕ, 1 ≤ n ∧ ‖hopping (cutWave q n) - freeEnergy q • cutWave q n‖ ≤ δ * ‖cutWave q n‖ := by obtain ⟨k, hk⟩ := exists_nat_gt (512 / δ^2) let n := k + 1 have hn : 1 ≤ n := by dsimp [n]; omega refine ⟨n, hn, ?_⟩ rw [cutWave_norm hq] have hδ2 : 0 < δ^2 := sq_pos_of_pos hδ have hk' : 512 < (k:ℝ) * δ^2 := (div_lt_iff₀ hδ2).mp hk have hb := free_error_norm_sq hq n hn have hn0 : 0 < (n:ℝ)+1 := by positivity have hn1 : (k:ℝ) < (n:ℝ)+1 := by dsimp [n]; push_cast; linarith have hx : 512 ≤ δ^2 * ((n:ℝ)+1) := by nlinarith have hy := mul_le_mul_of_nonneg_right hx hn0.le have hz : 0 ≤ δ * ((n:ℝ)+1) := by positivity nlinarith [norm_nonneg (hopping (cutWave q n) - freeEnergy q • cutWave q n)] theorem diagonal_error_bound (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) {q : ℂ} (hq : ‖q‖ = 1) (n : ℕ) (w ε : ℝ) (hε : 0 ≤ ε) (hclose : ∀ x ∈ cube n, |v x - w| ≤ ε) : ‖potential v hh hv (cutWave q n) - (w:ℂ) • cutWave q n‖ ≤ ε * ((n:ℝ)+1) := by let u := potential v hh hv (cutWave q n) - (w:ℂ) • cutWave q n have hu (x : Site) : u x = ((v x - w : ℝ):ℂ) * cutWave q n x := by simp only [u, lp.coeFn_sub, Pi.sub_apply, potential_apply, lp.coeFn_smul, Pi.smul_apply, smul_eq_mul, Complex.ofReal_sub] ring have hout : ∀ x ∉ cube n, u x = 0 := by intro x hx simp [hu, hx] have hsq : ‖u‖^2 ≤ ε^2 * ((n:ℝ)+1)^2 := by rw [finite_norm_sq u (cube n) hout] calc _ ≤ ∑ _x ∈ cube n, ε^2 := by apply Finset.sum_le_sum intro x hx have hp : ‖u x‖ ≤ ε := by rw [hu, norm_mul, cutWave_apply, ite_eq_left hx, plane_norm hq, mul_one, Complex.norm_real, Real.norm_eq_abs] exact hclose x hx nlinarith [norm_nonneg (u x)] _ = _ := by simp [cube_card]; ring change ‖u‖ ≤ _ nlinarith [norm_nonneg u, (show 0 ≤ ε * ((n:ℝ)+1) by positivity)] theorem anderson_shift (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) (u : Hilbert) (t : Site) : anderson v hh hv (permute (translation (-t)) u) = permute (translation (-t)) (anderson (fun x => v (x+t)) hh (fun x => hv (x+t)) u) := by ext x simp [anderson, hopping_apply, permute_apply, translation_apply, Prod.add_def, sub_eq_add_neg, add_comm, add_left_comm, add_assoc] theorem constant_box_relative (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) {q : ℂ} (hq : ‖q‖ = 1) (n : ℕ) (w : ℝ) {δ : ℝ} (hδ : 0 < δ) (t : Site) (hfree : ‖hopping (cutWave q n) - freeEnergy q • cutWave q n‖ ≤ δ * ‖cutWave q n‖) (hclose : ∀ x ∈ cube n, |v (x+t) - w| ≤ δ) : ∃ u : Hilbert, 0 < ‖u‖ ∧ ‖anderson v hh hv u - (freeEnergy q + (w:ℂ)) • u‖ ≤ (2*δ) * ‖u‖ := by let P := permute (translation (-t)) let u := P (cutWave q n) have hun : ‖u‖ = (n:ℝ)+1 := by dsimp [u]; rw [P.norm_map, cutWave_norm hq] refine ⟨u, by rw [hun]; positivity, ?_⟩ have he : anderson v hh hv u - (freeEnergy q + (w:ℂ)) • u = P ((hopping (cutWave q n) - freeEnergy q • cutWave q n) + (potential (fun x => v (x+t)) hh (fun x => hv (x+t)) (cutWave q n) - (w:ℂ) • cutWave q n)) := by dsimp only [u] rw [anderson_shift] simp only [P, map_add, map_sub, map_smul, anderson, add_apply, add_smul] abel rw [he, P.norm_map, hun] have hb := diagonal_error_bound (fun x => v (x+t)) hh (fun x => hv (x+t)) hq n w δ hδ.le hclose rw [cutWave_norm hq] at hfree exact (norm_add_le _ _).trans (by linarith) theorem spectrum_of_approximate (H : Hilbert →L[ℂ] Hilbert) (z : ℂ) (hap : ∀ ε : ℝ, 0 < ε → ∃ u : Hilbert, 0 < ‖u‖ ∧ ‖H u - z • u‖ ≤ ε * ‖u‖) : z ∈ spectrum ℂ H := by apply spectrum.mem_iff.mpr intro hi let B : Hilbert →L[ℂ] Hilbert := ↑hi.unit⁻¹ have hB : B * (algebraMap ℂ (Hilbert →L[ℂ] Hilbert) z - H) = 1 := by exact hi.val_inv_mul have hrec (u : Hilbert) : B (z • u - H u) = u := by have he := congrArg (fun A : Hilbert →L[ℂ] Hilbert => A u) hB simpa only [mul_apply_eq_comp, sub_apply, ContinuousLinearMap.algebraMap_apply, one_apply_eq_self] using he let C : ℝ := ‖B‖ + 1 have hC : 0 < C := by dsimp [C]; positivity let ε : ℝ := 1 / (2 * C) have hε : 0 < ε := by dsimp [ε]; positivity have hmul : C * ε = 1/2 := by dsimp [ε]; field_simp obtain ⟨u, hu, he⟩ := hap ε hε have hb : ‖u‖ ≤ ‖B‖ * ‖H u - z • u‖ := by calc _ = ‖B (z • u - H u)‖ := congrArg norm (hrec u).symm _ ≤ ‖B‖ * ‖z • u - H u‖ := B.le_opNorm _ _ = _ := by rw [norm_sub_rev] have hc : ‖B‖ * ‖H u - z • u‖ ≤ C * (ε * ‖u‖) := mul_le_mul (by dsimp [C]; linarith) he (norm_nonneg _) hC.le rw [← mul_assoc, hmul] at hc linarith theorem freeEnergy_surjective {e : ℝ} (he : e ∈ Set.Icc (-4:ℝ) 4) : ∃ q : ℂ, ‖q‖ = 1 ∧ freeEnergy q = (e:ℂ) := by have ht : e/4 ∈ Set.range Real.cos := by rw [Real.range_cos] constructor <;> linarith [he.1, he.2] obtain ⟨k, hk⟩ := ht refine ⟨Complex.exp ((k:ℂ)*Complex.I), Complex.norm_exp_ofReal_mul_I k, ?_⟩ dsimp [freeEnergy] rw [← Complex.exp_neg] have hneg : -((k:ℂ)*Complex.I) = ((-k:ℝ):ℂ)*Complex.I := by push_cast; ring rw [hneg, Complex.exp_ofReal_mul_I, Complex.exp_ofReal_mul_I] simp only [Real.cos_neg, Real.sin_neg, Complex.ofReal_neg] rw [hk] push_cast ring theorem iid_exists_ae {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ] {s : Set X} (hs : MeasurableSet s) (hp : μ s ≠ 0) : ∀ᵐ ω ∂Measure.infinitePi (fun _ : ℕ => μ), ∃ n, ω n ∈ s := by rw [ae_iff] have hn (n : ℕ) : Measure.infinitePi (fun _ : ℕ => μ) {ω | ¬ ∃ n, ω n ∈ s} ≤ (μ sᶜ) ^ n := by calc _ ≤ Measure.infinitePi (fun _ : ℕ => μ) (Set.pi (Finset.range n) (fun _ => sᶜ)) := measure_mono (by intro ω hω i hi exact fun his => hω ⟨i, his⟩) _ = (μ sᶜ) ^ n := by rw [Measure.infinitePi_pi _ (fun _ _ => hs.compl)] simp have hlt : μ sᶜ < 1 := by rw [measure_compl hs (measure_ne_top μ s), measure_univ] exact ENNReal.sub_lt_self (by simp) (by simp) hp exact le_antisymm (le_of_tendsto_of_tendsto' tendsto_const_nhds (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hlt) hn) zero_le abbrev Patch (n : ℕ) := Fin (n + 1) × Fin (n + 1) def patchSite (n : ℕ) (p : Patch n) : Site := ((p.1.val : ℤ), (p.2.val : ℤ)) def trialSite (n : ℕ) (k : ℕ) (p : Patch n) : Site := (((k * (n + 1) + p.1.val : ℕ) : ℤ), (p.2.val : ℤ)) theorem trialSite_injective (n : ℕ) : Function.Injective (fun kp : ℕ × Patch n => trialSite n kp.1 kp.2) := by rintro ⟨k, a, b⟩ ⟨l, c, d⟩ he have h1 : k * (n + 1) + a.val = l * (n + 1) + c.val := by have ht := congrArg Prod.fst he dsimp [trialSite] at ht exact_mod_cast ht have h2 : b.val = d.val := by have ht := congrArg Prod.snd he dsimp [trialSite] at ht exact_mod_cast ht have hk : k = l := by have hh := congrArg (fun z => z / (n + 1)) h1 rw [Nat.mul_comm k, Nat.mul_comm l, Nat.mul_add_div (Nat.succ_pos _), Nat.mul_add_div (Nat.succ_pos _)] at hh simpa only [Nat.div_eq_of_lt a.isLt, Nat.div_eq_of_lt c.isLt, Nat.add_zero] using hh have ha : a = c := by apply Fin.ext rw [hk] at h1 exact Nat.add_left_cancel h1 have hb : b = d := Fin.ext h2 simp [hk, ha, hb] theorem trial_law {h : ℝ} (hh : 0 < h) (n : ℕ) : (disorderLaw h).map (fun v k p => v (trialSite n k p)) = Measure.infinitePi (fun _ : ℕ => Measure.infinitePi (fun _ : Patch n => ProbabilityTheory.cond volume (Set.Icc (-h) h))) := by let := uniform_probability hh have hm := Measure.map_infinitePi_infinitePi_of_inj (P := fun _ : Site => ProbabilityTheory.cond volume (Set.Icc (-h) h)) (trialSite_injective n) have hc := Measure.infinitePi_map_curry (fun (_ : ℕ) (_ : Patch n) => ProbabilityTheory.cond volume (Set.Icc (-h) h)) rw [← hc, ← hm] rw [Measure.map_map (by fun_prop) (by fun_prop)] rfl theorem uniform_interval_pos {h w ε : ℝ} (hw : w ∈ Set.Ioo (-h) h) (hε : 0 < ε) : 0 < (ProbabilityTheory.cond volume (Set.Icc (-h) h)) (Set.Ioo (w - ε) (w + ε)) := by rw [ProbabilityTheory.cond_apply measurableSet_Icc] apply ENNReal.mul_pos · rw [ENNReal.inv_ne_zero, Real.volume_Icc] exact ENNReal.ofReal_ne_top · apply ne_of_gt apply lt_of_lt_of_le ((isOpen_Ioo.inter isOpen_Ioo).measure_pos volume (show (Set.Ioo (-h) h ∩ Set.Ioo (w - ε) (w + ε)).Nonempty from ⟨w, hw, by constructor <;> linarith⟩)) apply measure_mono exact Set.inter_subset_inter_left _ Set.Ioo_subset_Icc_self theorem constant_trial_ae {h w ε : ℝ} (hh : 0 < h) (hw : w ∈ Set.Ioo (-h) h) (hε : 0 < ε) (n : ℕ) : ∀ᵐ v ∂disorderLaw h, ∃ k : ℕ, ∀ p : Patch n, |v (trialSite n k p) - w| < ε := by let := uniform_probability hh let μ := ProbabilityTheory.cond volume (Set.Icc (-h) h) let s : Set (Patch n → ℝ) := Set.univ.pi (fun _ => Set.Ioo (w - ε) (w + ε)) have hs : MeasurableSet s := MeasurableSet.pi (Set.to_countable _) (by measurability) have hp : (Measure.infinitePi (fun _ : Patch n => μ)) s ≠ 0 := by dsimp [s] rw [Measure.infinitePi_pi_univ _ (fun _ => measurableSet_Ioo), tprod_fintype] exact Finset.prod_ne_zero_iff.mpr (fun _ _ => ne_of_gt (uniform_interval_pos hw hε)) have he := iid_exists_ae (Measure.infinitePi (fun _ : Patch n => μ)) hs hp have hm : MeasurePreserving (fun v k p => v (trialSite n k p)) (disorderLaw h) (Measure.infinitePi (fun _ : ℕ => Measure.infinitePi (fun _ : Patch n => μ))) := ⟨by fun_prop, trial_law hh n⟩ filter_upwards [hm.quasiMeasurePreserving.ae he] with v hv obtain ⟨k, hk⟩ := hv refine ⟨k, fun p => ?_⟩ have hp := hk p (Set.mem_univ p) exact abs_lt.mpr ⟨by linarith [hp.1], by linarith [hp.2]⟩ def HasConstantBoxes (v : Configuration) (h : ℝ) : Prop := ∀ w ∈ Set.Ioo (-h) h, ∀ ε : ℝ, 0 < ε → ∀ n : ℕ, ∃ t : Site, ∀ p : Patch n, |v (patchSite n p + t) - w| < ε theorem constant_boxes_ae {h : ℝ} (hh : 0 < h) : ∀ᵐ v ∂disorderLaw h, HasConstantBoxes v h := by have he : ∀ᵐ v ∂disorderLaw h, ∀ (q : ℚ) (l n : ℕ), (q : ℝ) ∈ Set.Ioo (-h) h → ∃ k : ℕ, ∀ p : Patch n, |v (trialSite n k p) - (q : ℝ)| < 1 / ((l : ℝ) + 1) := by simp only [ae_all_iff] intro q l n by_cases hq : (q : ℝ) ∈ Set.Ioo (-h) h · intro _ exact constant_trial_ae hh hq (by positivity) n · simp [hq] filter_upwards [he] with v hv intro w hw ε hε n let δ := min (ε / 2) (min (w + h) (h - w)) have hδ : 0 < δ := by dsimp [δ]; exact lt_min (by linarith) (lt_min (by linarith [hw.1]) (by linarith [hw.2])) obtain ⟨q, hq1, hq2⟩ := exists_rat_btwn (show w - δ < w + δ by linarith) have hd1 : δ ≤ ε / 2 := min_le_left _ _ have hd2 : δ ≤ w + h := (min_le_right _ _).trans (min_le_left _ _) have hd3 : δ ≤ h - w := (min_le_right _ _).trans (min_le_right _ _) have hq : (q : ℝ) ∈ Set.Ioo (-h) h := ⟨by linarith, by linarith⟩ have hwq : |(q : ℝ) - w| < ε / 2 := abs_lt.mpr ⟨by linarith, by linarith⟩ obtain ⟨l, hl⟩ := exists_nat_one_div_lt (show 0 < ε / 2 by linarith) obtain ⟨k, hk⟩ := hv q l n hq refine ⟨(((k * (n + 1) : ℕ) : ℤ), 0), fun p => ?_⟩ have hsite : patchSite n p + (((k * (n + 1) : ℕ) : ℤ), 0) = trialSite n k p := by ext <;> simp [patchSite, trialSite, Nat.cast_add, add_comm] rw [hsite] calc |v (trialSite n k p) - w| = |(v (trialSite n k p) - (q : ℝ)) + ((q : ℝ) - w)| := by congr 1; ring _ ≤ |v (trialSite n k p) - (q : ℝ)| + |(q : ℝ) - w| := abs_add_le _ _ _ < ε := by linarith [hk p] theorem cube_is_patch (n : ℕ) (x : Site) (hx : x ∈ cube n) : ∃ p : Patch n, patchSite n p = x := by simp only [cube, Finset.mem_product, Finset.mem_Icc] at hx refine ⟨(⟨x.1.toNat, by omega⟩, ⟨x.2.toNat, by omega⟩), ?_⟩ ext <;> simp only [patchSite, Int.toNat_of_nonneg, hx.1.1, hx.2.1] theorem spectrum_contains_shifted_free (v : Configuration) {h : ℝ} (hh : 0 ≤ h) (hv : ∀ x, |v x| ≤ h) (hbox : HasConstantBoxes v h) {w : ℝ} (hw : w ∈ Set.Ioo (-h) h) {q : ℂ} (hq : ‖q‖ = 1) : freeEnergy q + (w:ℂ) ∈ spectrum ℂ (anderson v hh hv) := by apply spectrum_of_approximate intro ε hε have hd : 0 < ε/2 := by linarith obtain ⟨n, hn, hfree⟩ := free_error_relative hq hd obtain ⟨t, ht⟩ := hbox w hw (ε/2) hd n have hc : ∀ x ∈ cube n, |v (x+t) - w| ≤ ε/2 := by intro x hx obtain ⟨p, rfl⟩ := cube_is_patch n x hx exact (ht p).le obtain ⟨u, hu, he⟩ := constant_box_relative v hh hv hq n w hd t hfree hc exact ⟨u, hu, by convert he using 1; ring⟩ theorem spectrum_contains_interior (v : Configuration) {h : ℝ} (hh : 0 < h) (hv : ∀ x, |v x| ≤ h) (hbox : HasConstantBoxes v h) {e : ℝ} (he : e ∈ Set.Ioo (-4-h) (4+h)) : (e:ℂ) ∈ spectrum ℂ (anderson v hh.le hv) := by let a := e/(4+h) have hp : 0 < 4+h := by linarith have hlo : -1 < a := by change -1 < e/(4+h); apply (lt_div_iff₀ hp).mpr; linarith [he.1] have hhi : a < 1 := by change e/(4+h) < 1; apply (div_lt_iff₀ hp).mpr; linarith [he.2] have hw : h*a ∈ Set.Ioo (-h) h := by constructor · nlinarith · nlinarith have hf : 4*a ∈ Set.Icc (-4:ℝ) 4 := ⟨by linarith, by linarith⟩ obtain ⟨q, hq, hqe⟩ := freeEnergy_surjective hf have hm := spectrum_contains_shifted_free v hh.le hv hbox hw hq have hea : (4+h)*a = e := by dsimp [a]; field_simp have hec : freeEnergy q + ((h*a:ℝ):ℂ) = (e:ℂ) := by rw [hqe, ← Complex.ofReal_add] congr 1 nlinarith rwa [hec] at hm theorem spectrum_eq_interval (v : Configuration) {h : ℝ} (hh : 0 < h) (hv : ∀ x, |v x| ≤ h) (hbox : HasConstantBoxes v h) : spectrum ℂ (anderson v hh.le hv) = spectralInterval h := by apply Set.Subset.antisymm (spectrum_subset v hh.le hv) have hi : Set.Ioo (-4-h) (4+h) ⊆ Complex.ofReal ⁻¹' spectrum ℂ (anderson v hh.le hv) := by intro e he exact spectrum_contains_interior v hh hv hbox he have hc := closure_minimal hi ((spectrum.isClosed (𝕜 := ℂ) (anderson v hh.le hv)).preimage Complex.continuous_ofReal) rw [closure_Ioo (by linarith : -4-h ≠ 4+h)] at hc intro z hz have hm := hc ⟨hz.2.1, hz.2.2⟩ have hez : (z.re:ℂ) = z := by apply Complex.ext <;> simp [hz.1] simpa only [Set.mem_preimage, hez] using hm theorem anderson_spectrum_ae {h : ℝ} (hh : 0 < h) : ∀ᵐ v ∂disorderLaw h, ∃ H : Hilbert →L[ℂ] Hilbert, IsAndersonOperator v H ∧ IsSelfAdjoint H ∧ spectrum ℂ H = spectralInterval h := by filter_upwards [bounded_ae hh, constant_boxes_ae hh] with v hv hbox exact ⟨anderson v hh.le hv, anderson_formula v hh.le hv, anderson_selfAdjoint v hh.le hv, spectrum_eq_interval v hh hv hbox⟩ theorem purePoint_of_completeOrthonormalEigenbasis {H : Hilbert →L[ℂ] Hilbert} (hb : HasCompleteOrthonormalEigenbasis H) : PurePoint H := by obtain ⟨b, hb⟩ := hb apply top_unique rw [← b.dense_span] apply Submodule.topologicalClosure_mono apply Submodule.span_mono rintro u ⟨k, rfl⟩ obtain ⟨z, hz⟩ := hb k exact ⟨(z : ℂ), b.orthonormal.ne_zero k, hz⟩ theorem eigenvalues_subset_spectrum (H : Hilbert →L[ℂ] Hilbert) : eigenvalues H ⊆ spectrum ℂ H := by rintro z ⟨u, hu, he⟩ apply spectrum_of_approximate intro ε hε refine ⟨u, norm_pos_iff.mpr hu, ?_⟩ rw [he, sub_self, norm_zero] positivity theorem closure_eigenvalues_subset_spectrum (H : Hilbert →L[ℂ] Hilbert) : closure (eigenvalues H) ⊆ spectrum ℂ H := closure_minimal (eigenvalues_subset_spectrum H) (spectrum.isClosed H) theorem spectrum_eq_closure_eigenvalues_of_completeOrthonormalEigenbasis {H : Hilbert →L[ℂ] Hilbert} (hb : HasCompleteOrthonormalEigenbasis H) : spectrum ℂ H = closure (eigenvalues H) := by classical obtain ⟨b, hb⟩ := hb choose eig heig using hb refine Set.Subset.antisymm ?_ (closure_eigenvalues_subset_spectrum H) intro z hz by_contra hzc rw [Metric.mem_closure_iff] at hzc push Not at hzc obtain ⟨ε, hε, hdist⟩ := hzc have hsep (k : ℕ) : ε ≤ ‖z - (eig k : ℂ)‖ := by simpa only [dist_eq_norm] using hdist (eig k : ℂ) ⟨b k, b.orthonormal.ne_zero k, heig k⟩ have hnz (k : ℕ) : z - (eig k : ℂ) ≠ 0 := norm_pos_iff.mp (hε.trans_le (hsep k)) have hbound (k : ℕ) : ‖(z - (eig k : ℂ))⁻¹ • ContinuousLinearMap.id ℂ ℂ‖ ≤ ε⁻¹ := by simp only [norm_smul, ContinuousLinearMap.norm_id, mul_one, norm_inv] exact inv_anti₀ hε (hsep k) let D : lp (fun _ : ℕ => ℂ) 2 →L[ℂ] lp (fun _ : ℕ => ℂ) 2 := lp.mapCLM 2 (fun k => (z - (eig k : ℂ))⁻¹ • ContinuousLinearMap.id ℂ ℂ) (inv_nonneg.mpr hε.le) hbound have hD (k : ℕ) : D (lp.single 2 k (1 : ℂ)) = (z - (eig k : ℂ))⁻¹ • lp.single 2 k (1 : ℂ) := by ext j change (z - (eig j : ℂ))⁻¹ * (lp.single (E := fun _ : ℕ => ℂ) 2 k (1 : ℂ) j) = (z - (eig k : ℂ))⁻¹ * (lp.single (E := fun _ : ℕ => ℂ) 2 k (1 : ℂ) j) by_cases h : j = k · subst j rfl · simp [lp.single_apply, h] let K : Hilbert →L[ℂ] Hilbert := b.repr.symm.toContinuousLinearEquiv.toContinuousLinearMap ∘L D ∘L b.repr.toContinuousLinearEquiv.toContinuousLinearMap have hK (k : ℕ) : K (b k) = (z - (eig k : ℂ))⁻¹ • b k := by change b.repr.symm (D (b.repr (b k))) = _ rw [b.repr_self, hD, map_smul, b.repr_symm_single] let A := algebraMap ℂ (Hilbert →L[ℂ] Hilbert) z - H have hA (k : ℕ) : A (b k) = (z - (eig k : ℂ)) • b k := by change z • b k - H (b k) = _ rw [heig k, sub_smul] have hdense : Dense (Submodule.span ℂ (Set.range b) : Set Hilbert) := Submodule.dense_iff_topologicalClosure_eq_top.mpr b.dense_span have hAK : A * K = 1 := by apply ContinuousLinearMap.ext_on hdense rintro u ⟨k, rfl⟩ change A (K (b k)) = b k rw [hK, map_smul, hA, smul_smul, inv_mul_cancel₀ (hnz k), one_smul] have hKA : K * A = 1 := by apply ContinuousLinearMap.ext_on hdense rintro u ⟨k, rfl⟩ change K (A (b k)) = b k rw [hA, map_smul, hK, smul_smul, mul_inv_cancel₀ (hnz k), one_smul] have hunit : IsUnit A := ⟨⟨A, K, hAK, hKA⟩, rfl⟩ exact spectrum.mem_iff.mp hz hunit end PlanarAnderson end end OAI