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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Probability.ClassicalON.AnnulusResponse
namespace OAI
noncomputable section
open scoped BigOperators ComplexConjugate Topology
open Filter
namespace ClassicalON
namespace LatticeGraph
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))
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
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
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
end LatticeGraph
end ClassicalON
end
end OAI
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