| import OAI.Probability.ClassicalON.AnnulusResponse |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ComplexConjugate Topology |
| open Filter |
| namespace ClassicalON |
| namespace LatticeGraph |
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| def rotationCostConstant (β C : ℝ) : ℝ := |
| 4*((6*β^2+4*β*Real.sqrt β+β)*(48*(2*modeBlockConstant C)^4)+ |
| β*(576*C^2*(2*modeBlockConstant C)^2)) |
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| def transferCostConstant (β L D : ℝ) : ℝ := |
| β*(512*D*(2*L+D))+2*(β*512*D)*Real.sqrt (β*(512*L^2))+(β*512*D)^2 |
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| theorem theta_response_bound (G : LatticeGraph) (k : ℕ) (hk : 0<k) (hG : G.WithinDyadicBox k) |
| (β : ℝ) (hβ : 0 ≤ β) (b : G.edges → ℝ) (hb : ∀ e,0 ≤ b e ∧ b e ≤ β) |
| (P : G.vertices → Option (Spin 3)) (hP : G.BoundaryPins k 3 P) |
| (C L D : ℝ) (hC : 0 ≤ C) (hL : 0 ≤ L) (hD : 0 ≤ D) |
| (hv : ∀ m s z, |squareProfile m s z| ≤ C) |
| (hLip : ∀ m s z w, |squareProfile m s w-squareProfile m s z| ≤ C*‖w-z‖) |
| (he : ∀ e : G.edges, |G.thetaEdge k e| ≤ L/(2:ℝ)^k ∧ |
| |G.thetaEdge k e-G.profileEdge k e| ≤ D/((2:ℝ)^k)^3) : |
| -(G.spinSystem 3 b P).secondAxisResponse (G.thetaEdge k) (G.thetaEdge k) ≤ |
| rotationCostConstant β C/(k:ℝ)+transferCostConstant β L D/(2:ℝ)^k := by |
| let S := G.spinSystem 3 b P |
| let a := rotationCostConstant β C/(k:ℝ) |
| let N : ℝ := (2:ℝ)^k |
| have hN : 0<N := by dsimp [N]; positivity |
| have hN1 : 1≤N := one_le_pow₀ (by norm_num) |
| have ha : 0≤a := by dsimp [a,rotationCostConstant]; positivity |
| have hr : -(S.complexSecondResponse (fun e => (G.profileEdge k e:ℂ)) |
| (fun e => conj (G.profileEdge k e:ℂ))).re ≤ a := by |
| simp only [Complex.conj_ofReal,S.complexSecondResponse_ofReal,Complex.ofReal_re] |
| exact G.profile_response_bound k hk hG β hβ b hb P hP C hC hv hLip |
| have ht := S.response_transfer β a hβ ha hb |
| (fun e => (G.thetaEdge k e:ℂ)) (fun e => (G.profileEdge k e:ℂ)) hr |
| simp only [Complex.conj_ofReal,S.complexSecondResponse_ofReal,Complex.ofReal_re, |
| ← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs] at ht |
| obtain ⟨hs1,hs2,hs3⟩ := G.theta_sum_bounds k hG L D hL hD he |
| have hsq : Real.sqrt (β*∑ e, |G.thetaEdge k e|^2) ≤ Real.sqrt (β*(512*L^2)) := |
| Real.sqrt_le_sqrt (mul_le_mul_of_nonneg_left hs2 hβ) |
| have h1n : 0 ≤ β*∑ e, |G.thetaEdge k e-G.profileEdge k e| := by positivity |
| have hh : -(S.secondAxisResponse (G.thetaEdge k) (G.thetaEdge k)) ≤ a+ |
| β*(512*D*(2*L+D)/N^2)+2*(β*(512*D/N))*Real.sqrt (β*(512*L^2))+(β*(512*D/N))^2 := by |
| exact ht.trans (add_le_add (add_le_add (add_le_add le_rfl |
| (mul_le_mul_of_nonneg_left hs3 hβ)) |
| (mul_le_mul (mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hs1 hβ) (by norm_num)) |
| hsq (Real.sqrt_nonneg _) (by positivity))) |
| (pow_le_pow_left₀ h1n (mul_le_mul_of_nonneg_left hs1 hβ) 2)) |
| apply hh.trans |
| change a+β*(512*D*(2*L+D)/N^2)+2*(β*(512*D/N))*Real.sqrt (β*(512*L^2))+(β*(512*D/N))^2 ≤ a+transferCostConstant β L D/N |
| rw [add_assoc a,add_assoc a] |
| apply add_le_add le_rfl |
| have hN2 : N ≤ N^2 := by nlinarith |
| have hQ : 0 ≤ β*(512*D*(2*L+D))+(β*512*D)^2 := by positivity |
| calc _ = (β*(512*D*(2*L+D))+(β*512*D)^2)/N^2+ |
| (2*(β*512*D)*Real.sqrt (β*(512*L^2)))/N := by ring |
| _ ≤ (β*(512*D*(2*L+D))+(β*512*D)^2)/N+ |
| (2*(β*512*D)*Real.sqrt (β*(512*L^2)))/N := |
| add_le_add (div_le_div_of_nonneg_left hQ hN hN2) le_rfl |
| _ = _ := by unfold transferCostConstant; ring |
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| theorem annulus_response_small (β : ℝ) (hβ : 0 ≤ β) (ε : ℝ) (hε : 0<ε) : |
| ∃ k : ℕ, 4 ≤ k ∧ ∀ (G : LatticeGraph), G.WithinDyadicBox k → |
| ∀ (b : G.edges → ℝ), (∀ e,0 ≤ b e ∧ b e ≤ β) → |
| ∀ (P : G.vertices → Option (Spin 3)), G.BoundaryPins k 3 P → |
| -(G.spinSystem 3 b P).secondAxisResponse (G.thetaEdge k) (G.thetaEdge k) ≤ ε := by |
| obtain ⟨C,hC,hv,hLip⟩ := profiles_uniform_bound |
| obtain ⟨L,D,hL,hD,he⟩ := theta_edge_bounds |
| have ht1 : Tendsto (fun k : ℕ => rotationCostConstant β C/(k:ℝ)) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop |
| have ht2 : Tendsto (fun k : ℕ => transferCostConstant β L D/(2:ℝ)^k) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop (tendsto_pow_atTop_atTop_of_one_lt (by norm_num)) |
| have ht : Tendsto (fun k : ℕ => rotationCostConstant β C/(k:ℝ)+transferCostConstant β L D/(2:ℝ)^k) atTop (𝓝 0) := by |
| simpa only [add_zero] using ht1.add ht2 |
| obtain ⟨k,hk,hkε⟩ := ((eventually_ge_atTop 4).and (ht.eventually_lt_const hε)).exists |
| refine ⟨k,hk,?_⟩ |
| intro G hG b hb P hP |
| exact (G.theta_response_bound k (by omega) hG β hβ b hb P hP C L D (by linarith) hL hD hv hLip |
| (he G k hk)).trans hkε.le |
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| end LatticeGraph |
| end ClassicalON |
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| end |
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| end OAI |
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