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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 | import OAI.Probability.ClassicalON.PeriodicPartition
namespace OAI
noncomputable section
namespace ClassicalON.LatticeGraph
def thetaPotential (G : LatticeGraph) (k : ℕ) (x : G.vertices) : ℝ :=
squareTheta (scaledSite k x.val)
def rotatedBoundaryPins (G : LatticeGraph) (k : ℕ) (P : G.vertices → Option (Spin 3))
(t : ℝ) : G.vertices → Option (Spin 3) :=
fun x => (P x).map (spinRotation (SpinSystem.potentialRotation (G.thetaPotential k) t x))
theorem thetaPotential_binary_pins (G : LatticeGraph) (k : ℕ)
(P : G.vertices → Option (Spin 3)) (hP : G.BoundaryPins k 3 P) :
∀ x s, P x=some s → G.thetaPotential k x=0 ∨ G.thetaPotential k x=1 := by
intro x s hs
rcases hP x s hs with h | h
· exact Or.inr (squareTheta_one _ h.1 h.2)
· exact Or.inl (squareTheta_zero _ h)
theorem rotatedBoundaryPins_boundary (G : LatticeGraph) (k : ℕ)
(P : G.vertices → Option (Spin 3)) (hP : G.BoundaryPins k 3 P) (t : ℝ) :
G.BoundaryPins k 3 (G.rotatedBoundaryPins k P t) := by
intro x s hs
cases hp : P x with
| none => simp [rotatedBoundaryPins,hp] at hs
| some u => exact hP x u hp
theorem rotatePotential_spinSystem (G : LatticeGraph) (k : ℕ) (b : G.edges → ℝ)
(P : G.vertices → Option (Spin 3)) (t : ℝ) :
(G.spinSystem 3 b P).rotatePotential (G.thetaPotential k) t =
G.spinSystem 3 b (G.rotatedBoundaryPins k P t) := rfl
theorem annulus_partition_ratio (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 → ∀ x y : ℝ,
1-ε ≤ (G.spinSystem 3 b (G.rotatedBoundaryPins k P x)).Z (fun _ => 1) /
(G.spinSystem 3 b (G.rotatedBoundaryPins k P y)).Z (fun _ => 1) := by
let δ := 2*ε/(2*Real.pi)^2
have hδ : 0<δ := by dsimp [δ]; positivity
obtain ⟨k,hk,hR⟩ := annulus_response_small β hβ δ hδ
refine ⟨k,hk,?_⟩
intro G hG b hb P hP x y
have hp := G.thetaPotential_binary_pins k P hP
have hr (t : ℝ) : -((G.spinSystem 3 b P).rotatePotential (G.thetaPotential k) t).secondAxisResponse
((G.spinSystem 3 b P).differential (G.thetaPotential k))
((G.spinSystem 3 b P).differential (G.thetaPotential k)) ≤ δ := by
exact hR G hG b hb (G.rotatedBoundaryPins k P t) (G.rotatedBoundaryPins_boundary k P hP t)
have h := (G.spinSystem 3 b P).potential_partition_ratio (G.thetaPotential k) δ hδ.le hp hr x y
have he : δ*(2*Real.pi)^2/2=ε := by dsimp [δ]; field_simp
rw [he] at h
exact h
end ClassicalON.LatticeGraph
end
end OAI
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